Incremental selective decode-and-forward relaying for power line communication

Incrementa l Selecti v e Decode-and-F orw ard Relaying for Po wer Line Communication Ankit Dubey ∗ , Chinmoy Kundu † , T elex M. N. Ngatched ‡ , Octavia A. Dobre ‡ , and Ranjan K. Mallik § ∗ Departmen t of ECE, National Institute of T echnolo g y Goa, Farmagud i, Ponda, Goa 403 401, Ind ia † School of Electronics, Electrical Engineer ing and Comp uter Scie n ce, Queen’ s Univ e r sity Belfast, U.K. ‡ Departmen t of ECE, Faculty o f Engineer ing and Applied Science, Memo rial University , Canad a § Departmen t of Electrical Engin eering, Ind ian I nstitute of T ec h nology Delhi, Hauz Khas, Ne w Delhi 11001 6, Ind ia ∗ ankit.du bey@nitgoa.ac.in, † c.kund u@qub.ac.uk, ‡ tngatched @grenfell.mu n .ca, odobre@mun.ca , § rkmallik@ee.iitd. ernet.in Abstract —In this paper , an incremental selecti ve decode-and- fo rward (ISDF) relay strategy is proposed for power line com- munication (PLC) systems to improv e the spectral effi ci en cy . T radition al decode-and-forward (DF) relaying employs two time slots by usin g h alf- duplex relay s which signifi cantl y reduces th e spectral efficiency . Th e ISDF strategy utilizes the relay only if the direct li nk quality fails to attain a certain informa tion rate, thereby impro ving the spectral effi cien cy . Th e path gain is assumed to be log-normally di stributed with v ery h igh distance dependen t signal attenu ation. F urthermore, th e additive noise i s modeled as a Bernoulli-Gaussian process to incorporate th e effects of impulsive noise contents. Closed-form expressions for the outage probability and the f raction of times the relay is in use, and an approx imate closed-fo rm expression for t he aver age bit error rate (BER) ar e derive d for th e binary p hase-shift keying signalin g scheme. W e observ e that the fraction of times the r elay i s in use can be significantly re duced compar ed to th e traditional DF strategy . It is also observed that at high transmit power , the spectral efficiency increases while the a verage BER decreases with increase in the required rate. Index T erms —Bernoulli-Gaussian i mp ulsive noise, BER, ISDF , log-normal distribution, PLC, r elay , spectral effi ci en cy . I . I N T R O D U C T I O N V arious mo d ern concepts, such a s home au tomation, real- time energy mon ito ring system, and sma r t g r id rely pr imarily o n commun ication systems. Furth ermore, the power-line com mu- nication (PLC) is a key solution b y providing gr eater app liance- to-applian ce conn ecti vity [1], [ 2 ]. Although PLC is one of the p referred commu nication so- lutions for such applicatio n s, it has various challeng es. The commun ication signals transmitted throu gh power lines suffer from ad ditiv e distortion , which comp rises both backg round and impu lsive no ise [1 ]. In ad dition to additive distortio ns, commun ication signals are also affected by multiplicative dis- tortions. Cables used to carry high amplitude alter nating po wer signals at very low frequency (around 50 or 60 Hz) b ecome hostile when car rying low amplitude commu nication signals at very high frequen cy; henc e , communication signals und ergo a heavy distance-dep endent atten uation [1]. Fu r thermore , due to reflections fro m various termination s, multi-path pro pagation occurs and causes the receiv e d signal stren gth to fluctuate with time. In most cases, the en velop e of these fluctuations follows the log-no r mal distribution [3] –[5]. Thu s, for reliable lo ng- distance c o mmunicatio n, it is essential to mitigate the effects of add itive and mu ltiplicativ e distor tions. A well established technique of relay-ba sed commu nication is ther efore proposed for PLC [6], [7]. For multi-h op transmission, a distributed space-time coding technique is introduc e d in [6], while a cooper a tive coding for n arrowband PLC is pro posed in [7]. The average bit erro r r ate (BER) and outage pr o bability analysis using dec ode-and -forward (DF) r elay is studied in [8], omittin g the direc t transmission. Recently , in [ 9], the correlation among multi-hop channels has also been con sidered for closely-placed DF relay s; still, the d ir ect transmission is ig nored. Further, very recently , a class o f mach ine learning schemes, namely multi- armed b andit, is pr oposed to solve the relay selection p r oblem for dual-h o p transmission in [10]. These works make use of half-du plex relay s, which requ ires two time slots for the end- to -end co m munication . Th e time slots required for the end-to -end commun ication can be sig- nificantly improved b y usin g the increm ental r e la y ing strategy [11], thereb y impr oving the co mmunicatio n rate or sp e ctral efficiency . In incremental relaying, the relay is used only if the direct tr ansmission from source to destinatio n fails to achieve a req u ired information rate or equ i valently a certain signal-to- noise (SNR) threshold. In co njunction with DF relays, incremental relaying can be applied with selecti ve relaying called incremental selective DF (ISDF) relaying , whereb y the relay is used o nly when the direct transmission fails and also the source to relay link achieves the required info rmation rate. Though increme n tal o r ISDF relaying has been investigated in wireless systems (see, e .g., [11], [ 12] and ref erences ther e in), to the best of our knowledge, it has not been studied in PLC systems yet. Motiv ated by the above discussions, in this paper , the ISDF strategy is proposed to enhance the spectral ef ficiency o f PLC systems. The outa g e pr obability an d average BER perfor mance are ev aluated . The PLC channels are assumed to follow the log- normal distribution with high distance- depende n t attenuation, and th e add iti ve noise is assumed to f o llow the Bernoulli- Gaussian pro c ess. T o get in sight into the spectral efficiency , the fraction o f times the r elay is in u se is also d e r i ved. Our main contributions are : i) to propose ISDF relaying for PLC systems to increase spectr al efficiency , and ii) finding closed- Fig. 1. PLC system model. form expr essions fo r the o utage prob a bility a n d the fraction of tim e s the relay is in use, and an approximate closed-f orm expression for the average BER considerin g th e binar y phase- shift keying ( BPSK) signaling scheme. The r est of the pa p er is o rganized as follows. Section I I describes the system model, while closed-for m expression s for the outag e proba b ility an d the fraction of times the relay is in use, and an approxim ate closed-fo rm expression for the a verag e BER are deriv ed in Sections III, IV, and V, r espectiv ely . Section VI pr esents numerical and simu lation results, and Section VII provides concluding remarks. Notation: E [ · ] denote the expectation o f its argu ment over the random variable ( r .v .) X , Pr ( · ) is the probab ility of an e vent, P e ( · ) is the probability of bit err o r or BER, F X ( · ) represen t the cumu la tive distribution func tio n (CDF) of the r . v . X , and f X ( · ) is the correspond ing probability density function (PDF). I I . S Y S T E M M O D E L The PLC system, as shown in Fig. 1, consists of a sou r ce S , a destination D , an d an ISDF relay , R . S tries to com m unicate with D over a power cable with the he lp of R to increase spectral efficiency . A link between any two nodes is denoted by i ∈ { S D, S R , RD } wh ere S D , S R , an d R D re p resent the links between S - D , S - R , and R - D , respecti vely . The len g th o f the power cab le between any two nodes is denoted by d i , and d S R + d RD = d S D . In the first phase, S b r oadcasts a symbol with power P T S . A pr e defined rate R th bits/sec/Hz is assum e d for successful decodin g at D . If the direct transmission rate exceeds R th , S transmits a n ew symb ol in the secon d ph ase. Otherwise, R forwards the decod ed sym bol to D only if the S R link can gu a rantee a certain rate in the second phase. It is assumed that the total power , P T , is divided equally amon g S and R . A. Cha nnel Model The received symbol y i throug h the i th link is expressed as y i = p P R i h i s + z i , i ∈ { S D , S R, R D } , (1) where P R i is the received power , h i is the channel gain of the i th link, z i is the ad d iti ve noise sample at the receiver , and s is the unit p ower transmitted symbol. The received power P R i depend s o n the transmit power , length of the power ca b le, and path loss. The chann el gain multiplier h i is mod eled as an indepen d ently distributed log-norm al r .v . with PDF f h i ( v ) = 1 v q 2 π σ 2 h i exp − 1 2  ln v − µ h i σ h i  2 ! , v ≥ 0 , (2) where the pa rameters µ h i and σ h i are the m ean and the standard d e viation of the nor mal r .v . ln ( h i ) , respectively . The ℓ th moment of h i is giv en by E  h ℓ i  = exp ℓµ h i + ℓ 2 σ 2 h i 2 ! . (3) W e assume un it energy of th e channel gain, i.e., E [ h 2 i ] = 1 . According to (3), this implies µ h i = − σ 2 h i . The dB equiv alen t of th e rec e i ved power through the i th link, i ∈ { S D , S R, R D } , P R i , can be expressed as P R i (dB) = P T S (dB) − d i (km) × P L (dB/km) , (4) where P L (dB/km) denotes the distance-d ependent path loss factor . B. SNR Distribution The symbo ls tra n smitted throu gh p ower lines su ffer fro m impulsive no ise as well as back ground noise [1]. W e assume the Bernoulli-Gaussian mo del [13] wh ich is m ostly used [8] . Thus, the additive noise sample z i can be written as z i = z W i + z B i z I i , (5) where z W i and z I i represent the backgro und an d impulsive noise samples, respectiv ely , and z B i is a Bernoulli r .v . which equals 1 with prob a b ility p and 0 with probability (1 − p ) . Th e samples z W i and z I i are taken from the Gaussian distribution with m ean zero and variance σ 2 W and σ 2 I , r espectiv ely . As backgr o und and impulsive noises have d ifferent or igin, z W i , z I i , an d z B i are indep endent [14]. The refore, the n oise samp les z i are independent and id entically distributed (i.i.d. ) r .v .s, each with PDF [13] p z i ( ν ) = 2 X j =1 p j q 2 π σ 2 j exp − ν 2 2 σ 2 j ! , (6) where p 1 = 1 − p , p 2 = p , σ 2 1 = σ 2 W , σ 2 2 = σ 2 W + σ 2 I . The av erage noise power , N 0 i = E  z 2 i  , is given as N 0 i = E  z 2 W i  + E  z 2 B i  E  z 2 I i  = σ 2 W (1 + p η ) , (7) where η = σ 2 I σ 2 W represents the p ower r atio of impu lsi ve n oise to backgrou nd noise. As the chann el gain h i is log-no rmally distributed, the correspo n ding instantaneous SNR, γ i = P R i h 2 i N 0 i , is also log- normally distributed with PDF f γ i ( w ) = 1 w q 2 π σ 2 γ i exp − 1 2  ln w − µ γ i σ γ i  2 ! , w ≥ 0 , (8) and param e ters µ γ i = 2 µ h i + ln P R i N 0 i , σ γ i = 2 σ h i . The CDF of γ i is therefore given by F γ i ( w ) = P r[ γ i ≤ w ] = 1 − Q  ln w − µ γ i σ γ i  , w ≥ 0 , (9) where Q ( · ) denotes the Gaussian Q -fu nction. C. R equir ed SNR Thres hold As the chan nel is corr upted by backgr ound noise with probab ility p 1 = (1 − p ) , and backgrou nd and impulsiv e n oise with prob ability p 2 = p , the instantaneous channel capa c ity can be expressed as [15] C i ( γ i ) = 2 X j =1 p j log 2 (1 + α j γ i ) , (10) where α 1 = 1+ pη 2 and α 2 = 1+ pη 2(1+ η ) [2]. Theref ore, f o r th e successful detection of the signal fro m th e dir ect link, the approx imate SNR thresho ld that sho uld be m aintained at D correspo n ding to th e ra te req uirement R th can be ob tained fr om (10) as Γ S D ≈ α 1 − p 1 α 2 − p 2 2 R th . (11) T o maintain the same rate requiremen t at D through the ha lf - duplex relay ed p ath, the S R or RD link shou ld main tain twice the rate o f the S D link , and hence, the required SNR threshold is Γ S R = Γ RD ≈ α 1 − p 1 α 2 − p 2 2 2 R th . (12) I I I . O U TAG E P RO B A B I L I T Y The outag e probability is defined as th e probab ility that th e instantaneou s channe l cap acity falls below a prede fin ed rate. An ou tage ev ent would occur if any of th e following e vents happen s: i) th e tr ansmitted symbol can not be de te c ted b oth from the S D and S R link s, or ii) the S D link fails to detect the symbol, and, ev en if R is able to correctly forward it, RD link fails to deliver . Thus, th e o utage pro bability can be expressed mathematically by summing up the events i) and ii) as P o ( R th ) = Pr [ γ S D < Γ S D ] Pr [ γ S R < Γ S R ] + Pr [ γ S D < Γ S D ] Pr [ γ S R > Γ S R ] × Pr [ γ RD < Γ RD ] . (13) Finally , using (9) and after som e algebra, the outage prob ability can be expressed in closed-form as P o ( R th ) = Q  µ γ S D − ln (Γ S D ) σ γ S D  Q  µ γ S R − ln (Γ S R ) σ γ S R  + Q  µ γ S D − ln (Γ S D ) σ γ S D  Q  ln (Γ S R ) − µ γ S R σ γ S R  × Q  µ γ RD − ln (Γ RD ) σ γ RD  . (14) I V . R E L AY U S A G E The mo re the relay is used f or data transmission, th e poore r the spectral ef ficiency , and the more the ad ditional complexity and delay requ ired in data processing. Hen ce, the fraction of times the relay is in use is o f great interest and f or the ISDF strategy . Th is nu mber can be obtained by finding the proba b ility that the S D lin k fails whereas the S R link attains the req uired rate thresho ld and is expressed as N = Pr [ γ S D < Γ S D ] Pr [ γ S R > Γ S R ] = Q  µ γ S D − ln (Γ S D ) σ γ S D  Q  ln (Γ S R ) − µ γ S R σ γ S R  . (15) V . A V E R AG E B E R A bit error ca n occur e ith er in th e direct tran sm ission or in the relayed transmission to D , accord ing to the selecti ve relayin g technique assumed . A bit erro r in the dir e ct transmission can occur in two ways: i) if its SNR e xceeds the req uired threshold, or , ii) if its SNR does not exceed the required thresh old and the relayed transmission is not u sed. Now , a bit error in the relayed transmission can o ccur if only one of the links b etween S R or RD is in error when the S R link SNR exceeds the requ ired threshold. Th e average BER for binary signaling can be written by summing up the prob abilities of all the ab ove e vents as P e = E [ P e ( γ S D | γ S D ≥ Γ S D )] + Pr [ γ S R < Γ S R ] E [ P e ( γ S D | γ S D < Γ S D )] + Pr [ γ S D < Γ S D ] (1 − E [ P e ( γ S R | γ S R ≥ Γ S R )]) × E [ P e ( γ RD | γ S R ≥ Γ S R )] + Pr [ γ S D < Γ S D ] E [ P e ( γ S R | γ S R ≥ Γ S R )] × (1 − E [ P e ( γ RD | γ S R ≥ Γ S R )]) . (16) For th e equ iprobab le BPSK sign a ling scheme, the instanta- neous BER as a function of γ , P e ( γ ) 1 , is giv en as P e ( γ ) = 2 X j =1 p j Q  √ α j γ  . (17) Thus, to obtain a closed-form expression fo r th e average BER in (16), we need the expectation operation of an integral o f the type E [ P e ( γ | y 1 < γ ≤ y 2 )] = 2 X j =1 Z y 2 y 1 p j Q  √ α j y  f γ ( y ) d y . (18 ) As γ follows the log-nor m al distribution as given in (8), we can write (18) as E [ P e ( γ | y 1 < γ ≤ y 2 )] = 2 X j =1 Z y 2 y 1 p j Q  √ α j y  × 1 y q 2 π σ 2 γ exp − 1 2  ln ( y ) − µ γ σ γ  2 ! d y . (19) Using the transform ation ln( y ) = 2 t − ln( α j ) , (19) can be rewritten as E [ P e ( γ | y 1 < γ ≤ y 2 )] = 2 X j =1 Z ln( √ α j y 2 ) ln( √ α j y 1 ) p j Q (exp( t )) × 2 q 2 π σ 2 γ exp − 1 2  2 t − ln( α j ) − µ γ σ γ  2 ! d t. (20) It is difficult to ev aluate the above integral in closed-form, as it contains a function of the form Q (exp( t )) . Ther efore, we 1 The subscrip t i ∈ { S R, RD , S D } is dropped here onwa rd to expl ain the relati onship bet ween BER and S N R in general. 15 20 25 30 35 40 45 50 55 60 65 70 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 P T (dB) Outage prob abi lit y R th = 1 , d S D = 0 . 4 km (n umer ical) R th = 3 , d S D = 0 . 4 km (n umer ical) R th = 5 , d S D = 0 . 4 km (n umer ical) d S D = 0 . 4 km (s i m u lation ) R th = 1 , d S D = 0 . 8 km (n umer ical) R th = 3 , d S D = 0 . 8 km (n umer ical) R th = 5 , d S D = 0 . 8 km (n umer ical) d S D = 0 . 8 km (s i m u lation ) Fig. 2. Outage probab ility versus total transmit powe r with σ h i = 3 dB, P L = 60 dB/km, p = 0 . 1 , and η = 10 for diffe rent v alues of R th and d S D . propo se a novel ap proxim a tio n using the curve fitting tech nique to deal with such a functio n. The a p proxim a tion is giv e n as Q (exp( t )) ≈ M X m =1 a m exp −  t − b m c m  2 ! , (21) where a m , b m , and c m are fitting con stants. The numbe r of summation terms, M , depends on the region of interest and accuracy o f the fit. A suitable value of M and correspo nding a m , b m , and c m values are furth er discussed in Section VI. Using the approx imation in (21), th e integral in (20) can be ev aluated in appr oximate clo sed-form as E [ P e ( γ | y 1 < γ ≤ y 2 )] ≈ M X m =1 2 X j =1 Z ln( √ α j y 2 ) ln( √ α j y 1 ) p j a m exp −  t − b m c m  2 ! × 2 q 2 π σ 2 γ exp − 1 2  2 t − ln( α j ) − µ γ σ γ  2 ! d t = M X m =1 2 X j =1 2 p j a m σ γ √ 2 A m exp − C m,j −  B m,j A m  2 !! ×  Q  √ 2  A m ln( √ α j y 1 ) − B m,j A m  − Q  √ 2  A m ln( √ α j y 2 ) − B m,j A m  , (22) where A m = q 1 c 2 m + 2 σ 2 γ , B m,j = b m c 2 m + ln( α j )+ µ γ σ 2 γ , C m,j = b 2 m c 2 m + (ln( α j )+ µ γ ) 2 2 σ 2 γ . Finally , using (22), the a verage BER in (16) can be expressed in approxim ate closed-f o rm as in (2 3). 20 25 30 35 40 45 50 55 60 65 70 10 −4 10 −3 10 −2 10 −1 10 0 P T (dB) Ave rage BER R th = 1 , P L = 6 0 dB/ km ( numer ica l) R th = 5 , P L = 6 0 dB/ km ( numer ica l) P L = 6 0 dB /km (s imulation ) R th = 1 , P L = 8 0 dB/ km ( numer ica l) R th = 5 , P L = 8 0 dB/ km ( numer ica l) P L = 8 0 dB /km (s imulation ) Fig. 3. A verage BER v ersus tot al tra nsmit po wer wit h σ h i = 3 dB, d S D = 0 . 4 km, p = 0 . 1 , and η = 10 for dif ferent v alues of R th and P L . V I . R E S U L T S A N D D I S C U S S I O N S Numerical and simu lation re su lts are presented here to val- idate the perfo r mance analysis. Unless o therwise mention ed, the f ollowing parameter s are c o nsidered. S D is chosen as 40 0 m and 800 m, respectively , in consistence with a small PLC system environment [2]. Dependin g on the power distribution network, in g eneral, σ h i lies in betwee n 2 dB to 5 dB [4]. Here we a ssum e σ h i = 3 dB, ∀ i , where the con version from absolute scale to dB scale is given by σ h i ( dB ) = 10 σ h i / ln 1 0 . A high value o f σ h i indicates h igh fluc tuation in the received signal power [3], [4 ]. Th e d istance-depen dent path lo ss factor depend s upon the ty pe o f cable an d car rier f requency used for the transm ission, a n d ranges fro m 4 0 to 1 00 dB/km [16]. Hen ce, P L = 60 and 80 dB/km are chosen, r espectiv ely . The v alu es of the impulsive n oise param eters are p = 0 . 1 and η = 10 , following [9]. Fitting constants f or the approx imation in (21) are obtained from the curve fitting tool of MA TLAB with M = 7 , root mean squ ared error (RMSE) 0 . 00069 3 1 , a nd sum of squares du e to erro r (SSE) of 0 . 000 4708 . The parameters calculated from the curve fitting are giv e n in T able I. Fig. 2 sho ws the outage probability versus P T , for different values of R th and d S D . The numerica l curves are o btained using (14) and are fo und to agree well with the simu lation results, thus v a lid ating o ur outag e ana lysis. T o achie ve an outage probability of 10 − 3 with R th = 3 bits/sec/Hz, the PLC system with d S D = 0 . 4 km req uires P T = 48 dB, wh ereas with d S D = 0 . 8 , th e transmit power requiremen t increases to 65 dB. Thus, it can be con cluded that for a fixed R th and P T , the outage perfo r mance degrades with increasing d S D . Fig. 3 shows th e a verage BER versus P T for dif ferent R th and P L values. The numerical curves are obtain ed using the app roximate closed-form expression derived in (23). T he numerical results are also in ag reement with the simulation P e = M X m =1 2 X j =1 2 p j a m σ γ S D √ 2 A m,S D exp − C m,j,S D −  B m,j,S D A m,S D  2 !!  Q  √ 2  A m,S D ln( p α j Γ S D ) − B m,j,S D A m,S D  +  1 − Q  ln Γ S R − µ γ S R σ γ S R  M X m =1 2 X j =1 2 p j a m σ γ S D √ 2 A m,S D exp − C m,j,S D −  B m,j,S D A m,S D  2 !! ×  1 − Q  √ 2  A m,S D ln( p α j Γ S D ) − B m,j,S D A m,S D  +  1 − Q  ln Γ S D − µ γ S D σ γ S D  ×    1 − M X m =1 2 X j =1 2 p j a m σ γ S R √ 2 A m,S R exp − C m,j,S R −  B m,j,S R A m,S R  2 !! ! M X m =1 2 X j =1 2 p j a m σ γ RD √ 2 A m,RD × exp − C m,j,RD −  B m,j,RD A m,RD  2 !!  Q  ln Γ S R − µ γ S R σ γ S R  ! + M X m =1 2 X j =1 2 p j a m σ γ S R √ 2 A m,S R × exp − C m,j,S R −  B m,j,S R A m,S R  2 !! ! 1 − M X m =1 2 X j =1 2 p j a m σ γ RD √ 2 A m,RD exp − C m,j,RD −  B m,j,RD A m,RD  2 !! ×  Q  ln Γ S R − µ γ S R σ γ S R  !) . (23) T ABLE I P A R A M E T E R S I N ( 2 1 ) F RO M T H E C U RV E F I T T I N G F O R M = 7 . m a m b m c m 1 0.466 5 -5.37 2.174 2 -0.000 7029 -3.674 0.117 8 3 0.016 5 -3.141 0.000 4957 4 0.283 1 -2.998 1.458 5 0.211 3 -1.764 1.06 6 0.174 2 -0.842 5 0.837 7 0.079 86 -0.110 9 0.639 9 results, thus v alidating our analysis. In general, it is o bserved that the perfo r mance improves as P T increases and also for fixed P T and R th , the performan ce degrades with incr easing the distance - depende nt p ath loss. Fu rther , it is noticed that wh en P T is low , the perfor mances d egrades with increasing R th ; howe ver, when P T is high , the p erforman ce imp roves with the increase in R th . This is an in teresting observation as with th e increase in R th , intuitiv ely , the av e r age BER shou ld degrad e at all SNRs. When R th increases, th e average BER de c r eases at a lower P T as neither S D nor S R can overcome the incr eased SNR threshold at D a n d R , respectively . If P T is increased further, R can eventually overcome the requ ired SNR th reshold due to com paratively low p ath loss and incre a sed re c ei ved power at it, hen ce, this o bservation. Mo reover , at hig her v alues of P T the strategy tend s to follow the direct transmission, an d hence, the BER curves for v arious R th are parallel. In Fig. 4, the fraction of times R is in u se for transmission versus P T is plo tted using (15) for different R th values an d 10 20 30 40 50 60 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P T (dB) F raction o f tim es R is in use R th = 1 , d f = 0 . 1 R th = 3 , d f = 0 . 1 R th = 1 , d f = 0 . 5 R th = 3 , d f = 0 . 5 Fig. 4. F raction of times R is in use versus P T with σ h i = 3 dB, d S D = 0 . 4 km, P L = 60 dB/km, p = 0 . 1 , and η = 10 for dif ferent value s of R th and d f . relay p lacements ( d f ), wh ere d S R = d f d S D and 0 < d f < 1 . It is observed that for various R th and d f , th e curves are bell- shaped and n ever reach unity . As P T increases, in itially relay usage increases du e to improved S R link quality , later relay usage decreases d ue to better direct link q uality , and hence, the bell-shape. Thus, it can be conclu ded that the ISDF is spe ctrally efficient when compared to the tr a ditional DF relayin g, which uses the relay in each transmission. Next, it is o bserved that as d f increases at a given R th , the curves shift tow ar d s r ig ht and the m aximum fractio n o f times the r elay is in use also decreases. This can b e explained by the fact that as the length of S R link in creases, th e received SNR at the re lay decreases, which in turn reduces th e fraction of tim es th e relay is in u se. Further, we can ob serve that at a giv en d f and beyond a certain P T , the fraction o f times the relay is in use for higher R th = 3 becomes more than lower R th = 1 due to the b ell-shape. This means that the spectral ef ficiency d e creases wh e n R th increases at higher P T . This also justifies the crossovers of the average BER p lots for the same R th beyond certain P T in Fig. 3. T h us, although the spec tral efficiency decrea ses at higher P T when R th increases, interestingly the average BER imp roves. V I I . C O N C L U S I O N In this work, th e ISDF r elaying strate gy has been introduced for PLC systems to impr ove spectral efficiency . Closed-form expressions f or the outage probability and th e fraction of tim es the relay is in use alo ng with an app roximate closed-fo rm expression for the average BER are derived co nsidering th e BPSK signaling scheme. Log -normal fading an d Bernou lli- Gaussian impulsive n oise are consider ed for the ana ly sis. It is observed that a t lower transmit p ower , th e per f ormance degrades as th e req uired rate, p ath lo ss, or end-to-end distance increases. 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