MULTICARRIER transmission techniques have been proposed

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1 1 Performnce Anlysis of Anlog Inermienly Nonliner Filer in he Presence of Impulsive Noise Rez Brzideh, Blsubrmnim Nrjn, Alexei V. Nikiin, Solmz Niknm, Deprmen of Elecricl nd Compuer Engineering, Knss Se Universiy, Mnhn, KS, USA. Nonliner LLC, Wmego, KS 66547, USA. Emil:rezbrzideh, bl, rxiv: v1 [eess.sp] 1 Nov 18 Absrc An Adpive Nonliner Differenil Limier (ANDL) is proposed in his pper o efficienly llevie he impc of impulsive noise (IN) in communicion sysem. Unlike exising nonliner mehods, he ANDL is implemened in he nlog domin where he broder cquisiion bndwidh mkes ouliers more deecble nd consequenly i is esier o remove hem. While he proposed ANDL behves like liner filer when here is no oulier, i exhibis inermien nonlineriy in response o IN. Therefore, he srucure of he mched filer in he receiver is modified o compense he filering effec of he ANDL in he liner regime. In his pper, we qunify he performnce of he ANDL by deriving closed-form nlyicl bound for he verge signl-o-noise rio (SNR) he oupu of he filer. The clculion is bsed on he ide h he ANDL cn be perceived s ime-vrin liner filer whose bndwidh is modified bsed on he inensiy of he IN. In ddiion, by linerizing he filer ime prmeer vriions, we re he ANDL s se of liner filers where he exc opering filer given ime depends upon he mgniude of he ouliers. The heoreicl verge bi error re () is vlided hrough simulions nd he performnce gins relive o clssicl mehods such s blnking nd clipping re qunified. Index Terms Impulsive noise (IN), nlog nonliner filer, dpive nonliner differenil limier (ANDL), orhogonl frequency-division muliplexing (OFDM). I. INTRODUCTION MULTICARRIER rnsmission echniques hve been proposed o cope wih he frequency seleciviy of he propgion chnnel in mny pplicions [1]. Priculrly, orhogonl frequency-division muliplexing (OFDM) is widely used in mny pplicions in vehiculr communicions rnging from wired communicion such s Power-line communicion (PLC) in Home-Plug Green PHY sndrd for VG communicions [] o wireless communicions such s 8.11p Wireless Access in Vehiculr Environmens (WAVE) sndrd [3], nd underwer cousic (UWA) communicion [4]. However, OFDM provides some level of robusness gins impulsiviy, sysem performnce cn sill degrde if he impulsive noise (IN) exceeds cerin hreshold nd is effec ges spred over ll subcrriers [5]. Tking n OFDM-bsed sysem s n exmple, his pper inroduces nd nlyiclly qunifies he performnce of n nlog inermienly nonliner filer in he presence of IN. A. Reled work Mny echniques hve been explored in prior effors o miige he impc of IN. For exmple, robus ierive chnnel decoding echniques hve been used o meliore bi error re () in impulsive environmens [6], [7]. I hs been shown h coding echniques re mosly effecive in single crrier schemes nd here is no gin in OFDM sysems [8]. In ddiion, frequency or ime domin inerleving [9] [11] re no effecive in highly impulsive environmens. Moreover, compressive sensing (CS) echniques re used o esime IN by mesuremens on null subcrriers of OFDM [1] [15]. In [16] non-prmeric lgorihm is proposed by exension of [1] o sprse Byesin lerning (SBL) pproch [17]. A combinion of fcor-grph-bsed receiver nd messgepssing echnique [18] is proposed in [19] o miige IN. High mpliude nd shor durion of IN hs lso moived he use of vrious memoryless nonliner pproches such s clipping [], blnking [1], [], liner combinion of blnking nd clipping [3], deep clipping [4], nd muliplehreshold blnking/clipping [5]. The oupu signl-o-noise rio (SNR) cn be mximized by opimizing he hresholds used in he memoryless nonliner pproches. However, he clipping nd blnking hresholds re usully experimenlly derived. In [6], hreshold opimizion bsed on Neymn- Person crierion is proposed nd n nlyicl equion for he qusi-opiml blnking nd clipping hresholds is provided in [7]. Bndwidh reducion in he process of nlog-odigil conversion (ADC) is he min drwbck of ll hese digil nonliner pproches. Therefore, in our prior works we proposed using Blind Adpive Inermienly Nonliner Filers (BAINFs) o miige he IN before he ADC. An Adpive Nonliner Differenil Limier (ANDL) is considered s one relizion of BAINFs nd he bsics of ANDL re sudied in [8], [9]. A prcicl implemenion of BAINFs s Adpive Cnonicl Differenil Limier (ACDL) long wih mched filer modificion is discussed in [3] o miige he IN in PLC sysem in rel ime. B. Conribuions In his pper, simplified blind dpive inermienly nonliner filer rchiecure is proposed nd unique pproch o nlyse is performnce is inroduced. The min conribuions of his work cn be summrized s follows: Inroducing proper model for IN which cpures is chrcerisics in nlog domin while minining equivlency wih he common models used in discree domin. In order o reduce he complexiy of he nlyicl derivions, he proposed ANDL is simplified. However,

2 Informion Bis d d d... 1 Consellion Mpper (PSK or QAM) D/A s k S/P IFFT P/S s ( ) Pulse shping (Roo Rised Cosine) Therml Noise + Impulsive Noise r( ) dˆ dˆ ˆ 1d... Consellion Dempper & Deecion P/S FFT S/P Modified Mched Filer A/D ANDL Fig. 1: Sysem model block digrm. we show h his simplificion does no degrde he performnce of he proposed filer. The performnce of he ANDL is nlyiclly qunified by pproximing he ANDL s se of liner filers. Here, he exc liner filer h operes given ime depends upon he mgniude of he ouliers. Then, closed-form nlyicl bound is derived for he verge SNR he oupu of he proposed filer nd he nlyicl performnce is vlided by simulion. The improvemen in SNR nd is due o he fc h, unlike clssicl IN miigion mehods, ANDL is implemened in he nlog domin where he ouliers re sill brodbnd nd disinguishble. Disproporionl effec of he ANDL on he signl of ineres nd IN increses he SNR in he desired bndwidh by reducing he specrl densiy of he IN wihou significnly ffecing he desired signl. The heoreicl performnce of he ANDL is vlided vi simulion of n OFDM-bsed sysem in IN environmens. Moreover, we highligh he superioriy of our pproch over convenionl echniques such s blnking, clipping, nd liner filering. The reminder of his pper is orgnized s follows. Secion II describes he sysem nd noise models. Secion III presens he fundmenls of he ANDL long wih mched filer modificion nd resoluion prmeer clculion. Liner pproximion of he ANDL nd oupu SNR derivions re deiled in secion IV. Secion V presens heoreicl nd simulion resuls nd finlly conclusions re drwn in Secion VI. II. SYSTEM AND NOISE MODELS Fig. 1 illusres he block digrm of he considered OFDM-bsed sysem. Here, he moduled d s k is pssed hrough n inverse discree Fourier rnsform (IDFT) o genere OFDM symbols. A roo rised cosine (RRC) wveform wih roll-off fcor β is used o shpe nd rnsmi he OFDM signl hrough he chnnel. The rnsmied nlog signl envelope in ime domin cn be expressed s s() = 1 N 1 s k e j πk T p(), < < T, (1) N where N represens he number of subcrriers; p() denoes he pulse shpe, nd T is he cive OFDM symbol durion. Under perfec synchronizion, he received signl in n ddiive noise chnnel is given by r() = s()+w()+i(). () Here, s() denoes he desired signl wih vrince σs nd bndwidhb s ;w() is complex Gussin noise wih men zero nd vrince σw; nd i() represens he IN wih men zero nd vrince σi σ w. Wihou loss of generliy, since he min objecive of his pper is o demonsre novel pproch o miige IN, he effec of chnnel fding is elimined in (). According o he srucure of he receiver in Fig. 1, he proposed ANDL is implemened before he ADC s fron end filer nd he mched filer is modified o compense he filering effec of he ANDL in liner regime. In he following, we begin wih review of he impulse noise model. A. Impulsive Noise Model The widely used IN models ssume he presence or bsence of srong noise componen s he relizion of wo muully exclusive evens [31]. To nlyze nd evlue sysem performnce, we propose model h cpures chrcerisics of n IN in he nlog domin. The considered IN consiss of shor durion high powered impulses wih rndom rrivls nd corresponds o i() = ν() A k [θ( k ) θ( k τ s )]. (3) k=1 Here ν() represens complex whie Gussin noise process wih zero men; A k is he mpliude of k h pulse nd modeled by Gussin rndom vrible; k is rrivl ime of Poisson process wih prmeer λ; θ() denoes he Heviside uni sep funcion, nd τ s is he durion of IN. In generl he durion τ s cn chnge rndomly for ech burs bu here, for simpliciy, we ssume fixed verge durion for ll burss. However, i is imporn o noe h he mehod nd resuls presened in his work cn be esily exended o he cse when he IN durion is rndom. The resuling ime nd frequency domins represenion of his noise in nlog domin is depiced in Fig..

3 3 5 Ampliude.1. Time (ms) τ()/τ PSD Freq (MHz) x() χ() /α() Fig. 3: ANDL ime prmeer τ() = τ( x() χ() ). Fig. : Asynchronous impulsive noise Noe h, while (3) cpures bursy IN wih rndom mpliude in nlog domin, i lso cn represen Bernouli- Gussin IN model in ime durion T wih verge success probbiliy ε given by [ e λt (λt) k ε = kτ s ]/T k! [ ] e λt (λt) k 1 = λτ s (k 1)! k=1 [ ] e λt (λt) k = λτ s k! = λτ s. (4) In he nex secion, we discuss he design nd implemenion of ANDL in deil. III. FUNDAMENTALS OF ANDL An inroducion o he fundmenls of he ANDL nd finding n efficien vlue for he resoluion prmeer is provided in his secion. A. ANDL Design ANDL is blind dpive inermienly nonliner filer h, cn be perceived s firs order ime vrying liner filer. According o he bsic concep of he proposed ANDL [8], [9], he ime prmeer τ() vries proporionlly wih he mgniude of he difference beween inpu nd oupu of he filer. Therefore, we hve χ() = x() τ( x() χ() ) χ(), (5) where x() nd χ() re he inpu nd oupu of he filer, respecively, nd χ() denoes he firs ime derivive of χ(). As shown in Fig. 3, he ime prmeer τ() = τ( x() χ() ) is given by 1 x() χ() α() τ( x() χ() ) = τ x() χ() α() oherwise (6) where τ is fixed ime consn nd α() is he resoluion prmeer of he filer. The vlue of α() should be deermined properly in order o miige he IN efficienly. In generl, he, ANDL is n inermien nonliner filer nd behves linerly, when he mgniude of he difference signl x() χ() remins wihin cerin rnge deermined by he resoluion prmeer α(). This llows us o void insbiliies h re ofen ssocied wih nonliner filering. However, in cse of ouliers, he proper selecion of α() leds he ANDL o he nonliner regime o suppress he ouliers. Bsed on (6), ANDL is exremely ggressive owrd high mpliude IN, i.e., lrger spikes in he inpu signl will resul in greer suppression he oupu. I is worh noing h, we exend he works in [8], [9] by dding mched filer modificion module o compense for he ANDL in he liner regime. The impulse response h mod [k] of he modified mched filer in he discree domin cn be expressed s [3], h mod [k] = h[k]+τ ḣ[k], (7) where h[k] is he impulse response of he mched filer nd ḣ[k] denoes he firs ime derivive of h[k]. The modificion is done in he discree domin, s his reduces he compuionl complexiy nd neglecs he need for exr hrdwre componens. The compension of he modified mched filer on he performnce of n OFDM sysem wih B s = 1kHz nd binry phse shif keying (BPSK) modulion is shown in Fig. 4. A roo-rised-cosine filer wih roll-off fcor 1/4 is considered s mched filer in Fig. 4. Therefore, he performnce loss of ANDL in liner regime is compensed by modified mched filer when here is no IN. B. Resoluion Prmeer Clculion According o he srucure of ANDL, he objecive is o deermine ime-dependen resoluion prmeer α() h enhnces he quliy of non-sionry signls under imevrying noise condiions. Therefore, n efficien vlue of α() should llow o mximize he suppression of he IN wihou disoring he signl of ineres. I is ssumed h he power of herml noise is fixed over one OFDM symbol durion. Therefore, he resoluion prmeer is consn (α()=α) in he durion of ech OFDM symbol nd i only chnges cross symbols. A proper vlue of resoluion prmeer α cn be found bsed on difference signl x() χ() when here is no IN. An esime of he foremenioned difference signl cn be obined by pssing signl s()+w() hrough liner highpss filer. Le z() be given by differenil equion for

4 4 1-1 n Modified Mched Filer Mched Filer Theoreicl AWGN /N Fig. 4: Performnce comprison beween mched filer nd modified mched filer in he presence of ANDL for BPSK modulion. β =.5, τ =1/(4πB s). 1 3 Fig. 5: ANDL ime prmeer τ = τ(κ x ). n k x he firs order highpss filer wih he ime consn τ. Then, we hve [9] z() = τ [ṡ()+ẇ() ż()]. (8) As derived in our preliminry work [9], n efficien vlue of he resoluion prmeer α eff,ζ for (1 ζ) level disorionless filering of he rnsmied OFDM signl in herml noise is given by α eff,ζ erf 1 (1 ζ) σ z, (9) where σ z is he vrince of z(); erf(.) represens he error funcion; nd ζ is sufficienly smll consn (e.g., ζ = ). Now h we hve summrized he srucure nd operion of he ANDL, in he nex secion we derive nlyicl expressions for he verge SNR he ANDL oupu. IV. LINEAR APPROXIMATION OF THE ANDL In order o chrcerize he heoreicl performnce of he ANDL we employ liner pproximion. A. Time Prmeer τ() Approximion According o (6), he proposed ANDL eners he nonliner regime only he ime of incoming IN where he difference signl x() χ() would be pproximely equl o x(). Therefore, he ime prmeer of he ANDL in (6) cn be pproximed s 1 for κ x() α τ(κ x() ) = τ oherwise κ x() α, (1) whereα = α eff,ζ, ndκis posiive consn h cn be used o une he modified ANDL for vrious IN models. In order o find he heoreicl performnce we pproxime he ANDL by combinion ofnliner filers s illusred in Fig. 5. Here, he Primry ANDL Approximion, = 1 Approximion nd Linerizion, = 1, n = 1, =. Approximion nd Linerizion, = 1, n = 1, =. Approximion nd Linerizion, = 1, n = 5, = /N Fig. 6: Liner pproximion of ANDL. SIR = db, λ = B s, τ s = 1µs. ime consn of ech individul liner filer cn be expressed s τ, κ x() < α τ 1 = α1 α τ, α < κ x() < α 1 τ() =.. (11). τ k = α k α τ, α k 1 < κ x() < α k As cn be seen in Fig. 6, he performnce of he pproximed ANDL in (1) wih κ = 1 is lmos he sme s he primry ANDL in (6). Fig. 6 lso shows h he pproximion wih combinion of n liner filers resuls in performnce equivlen o he (6). Theoreiclly, we hve he bes pproximion when n where he difference beween wo consecuive filers α = α k α k 1, 1 k n is smll nd he vlues of α k re opimized. In his work, for simpliciy, he linerizion is performed ssuming consn α. Fig. 6 shows h in prcice, resonble vlue of n nd α h gurnee α n = α + n α > mx x() (cover he enire rnge of x() ) ensures he ccurcy of he pproximion. In our ANDL srucure, he received signl psses hrough brodbnd lowpss filer o limi he inpu noise power while ensuring h he IN is no excessively spred ou in ime

5 Ampliude 5 Considering sufficienly brodbnd fron end filer, he inpu signl x() for ANDL cn be represened by sionry mixure of wo Gussin componens weighed by 1 ε nd ε. Therefore, he probbiliy densiy funcion (PDF) of he inpu signl x() cn be expressed vi Gussin Mixure (GM) model given by where f X (x) = (1 ε)φ x1 (,σ 1)+εφ x (,σ ), (1) x 1 () = s()+w() N(,σ 1 = σ s +σ w) x () = s()+w()+i() N(,σ = σ s +σ w +σ i ), (13) nd φ x (.) is he Gussin PDF defined by φ x (µ,σ ) = 1 πσ e (x µ) σ. (14) Bsed on he GM model nd ccording o (11), he verge filering effec of he ANDL cn be compued vi n verged ime prmeer τ corresponding o E[τ] = (1 ε) p k,1 τ k +ε p k, τ k, (15) where, Pr( < κ x1 () < α p k,1 = ), k = Pr(α k 1 < κ x 1 () < α k ), k = 1,...,n ( ) 1 erfc κσ1 α, k = = ) erfc( ) erfc( αk 1 αk κσ1 κσ1,k = 1,...,n, nd Pr( < κ x () < α p k, = ), k = Pr(α k 1 < κ x () < α k ), k = 1,...,n ) 1 erfc( α κσ, k = = ) erfc( ) αk 1 erfc( αk κσ κσ,k = 1,...,n. Here, erfc(.) represens he complemenry error funcion. B. Oupu of he ANDL (16) (17) Considering (11), he ANDL cn be pproximed by weighed combinion of n liner filers wih ech of hem funcioning wih probbiliies corresponding o (16) nd (17). Thus, he verge oupu of he filer bsed on mixure model inpu cn be expressed s χ1 (), wih probbiliy 1 ε χ() =, (18) χ (), wih probbiliy ε where χ 1 () = n p k,1 [s()+w()] h k ()}, χ () = n p k, [s()+w()+i()] h k ()}. (19) Here, h k () is firs order liner lowpss filer wih ime consn τ k. In order o qunify he oupu power of ech D Time Cons. = Time Cons. = Fig. 7: Sep Response of he ANDL. individul filer, we consider squre pulses s n inpu (if no, ech shpe cn be pproximed by summion of nrrower squre pulses). According o Fig. 7, he oupu of he proposed ANDL consiss of wo prs y 1 () (red line) nd y () (green line) which re given by y 1 () (τ,) = (1 e τ), y () (τ,) = e ( ) τ,, () where τ is he ime prmeer for y 1 () (i.e., τ k in k h region of (11)); τ represens he ime consn nd i is deermined bsed on he bndwidh of desired signl; is durion of squre pulse wih mpliude, nd = (1 e τ ). Noe h τ = τ when here is no IN. Thus, given τ, τ nd, he corresponding oupu power fer lowpss filering for single pulse is given by P (τ,) = (P 1 +P ) (τ,) = y 1 d+ y d = (1 e τ ) d+ e ( ) τ d = [ τ e τ τ +τe τ 3 ] + τ. This moun of power is he ol residul power fer filering which consiss of power of he desired signl, herml, nd impulsive noises. In order o find heir individul conribuions, we use verge residul power for desired signl nd herml noise bu for IN we clcule he residul power for ech region in Fig. 5, seprely. Since he ANDL is pproximed by se of liner filers nd he mpliude vriion of he desired signl is much smller hn IN vriion (lower bndwidh), he verge residul power of desired signl cn be deermined by verging over τ nd, h is P s = E τ, [P τ, ] = P (τ,).f T (τ).f A ()dτd. (1) In he cse of he desired signl, rndom vrible corresponds o s() which hs folded-norml disribuion (s()

6 6 hs Gussin disribuion). Therefore, we hve ( P s = E [ s() ] (1 ε) p k,1 P (τk,1) where +ε p k, P ) (τk,1), () E[ s() ] = σ s π e( µ s /σ s ) +µ s (1 φ( µ s σ s )). (3) Similrly, in he cse of herml noise, he rndom vrible corresponds o w() nd we hve ( P w = E [ w() ] (1 ε) p k,1 P (τk,1) where +ε p k, P ) (τk,1), (4) E[ w() ] = σ w π e( µ w /σ w ) +µ w (1 φ( µ w σ w )). (5) The mpliude vriion of he IN is much lrger hn he mpliude vriion of he desired signl nd herml noise. However, i is possible h some IN my be buried wihin he desired signl nd herml noise. If h is he cse, hen here will be no wy o disinguish beween IN nd oher componens of he received signl in bnd limied sysem. This problem highlighs he dvnge of he proposed ANDL which is implemened in nlog domin where wide cquisiion bndwidh mkes he IN more disinguishble. Thus, he bsolue vlue of IN is more likely o be lrger hn he resoluion prmeer. Consequenly, he IN will encouner filer wih lrge τ proporionl o is mpliude s shown by dshed lines in Fig. 7. Therefore, we find he verge mpliude of IN in ech region of Fig. 5 nd for simpliciy we pick he cener of ech region excep in he firs region where α is picked s represenive of he mpliude of IN. Thus, we hve α, k = E[ i k ] = α + (k 1) α, k = 1,...,n, (6) nd he verge residul power of IN fer he linerized ANDL is given by P i = ε E [ i k ].p k,.p (τk,1). (7) Finlly, he verge oupu SNR cn be expressed s SNR vg = P s P w +P i. (8) Therefore, he verge cn be bounded using Jensen s inequliy. For exmple, for BPSK vg Q( SNR vg ) where Q(.) is he Q-funcion. V. SIMULATION RESULTS In his secion, he nlyicl resuls derived in he previous secions re vlided hrough simulions. In ddiion, SNR nd of n OFDM sysem wih BPSK modulion re used o compre he performnce of he proposed nlog nonliner filer o oher convenionl pproches such s liner filering, blnking nd clipping. As specific exmple, n OFDM-bsed sysem wih signl bndwidh B s = 1kHz nd N = 51 subcrriers is chosen s reference, bu he conclusions cn be exended o ny OFDM sysem s long s he number of subcrriers is lrge enough o sisfy he Gussin signl ssumpion. The sysem is invesiged in n ddiive noise environmen h consiss of wo componens: (i) herml noise, (ii) synchronous rndom IN wih normlly disribued mpliudes cpured by Poisson rrivl process wih prmeer λ nd ime durion τ s. To miige he IN, firs order ANDL wih τ =1/(4πB s ) is used. I is imporn o noe h when α he ANDL becomes firs order liner lowpss filer nd modified mched filer is used o llevie he filering effec of ANDL in he liner regime. To emule he nlog signls in he simulion, he digiizion re is chosen o be significnly higher (by bou wo orders of mgniude) hn he ADC smpling re. Noe h in ll simulions, (i) he opimum hresholds for blnking nd clipping re found bsed on n exhusive numericl serch, (ii) he resoluion prmeer α() for ANDL is deermined bsed on expression (9) wih low compuionl complexiy, nd (iii) κ = 1, α =., nd he number of qunizion levels n is deermined ccording o he dynmic rnge of incoming signl nd considered α. Fig. 8 shows he properies of he signl in ime nd frequency domin, nd is mpliude disribuion for differen mehods of IN miigion. In Fig. 8, he blck dshed lines (shded re) represen he desired signl (wihou noise), nd he colored solid lines represen he signl+noise mixures. The lefmos pnels show he ime domin rces, he righmos pnels show he power specrl densiy (PSDs), nd he middle pnels show he mpliude densiies (PDFs). From he pnels of he ls row, i is cler h he ANDL efficienly reduces he specrl densiy of he IN in he signl pssbnd wihou significnly ffecing he signl of ineres. By compring he pnels of row LIN (Liner), CLP (Clipping) nd BLN (Blnking) wih row ANDL (specilly PSDs pnels), i cn be seen h he chieved improvemen due o ANDL in he quliy of he bsebnd signl is significn. In he following, he foremenioned improvemen is shown in erms of SNR nd. The SNR performnce for liner filer, ANDL, blnking, nd clipping in vrious noise composiions is compred in Fig. 9. According o Fig. 9, ll pproches provide effecively equivlen performnce when herml noise domines he IN. However, he superioriy of he ANDL is highlighed when he IN is dominn nd in low SNR (SNR less hn zero) is performnce is lmos insensiive o furher increse in he IN power. The poency of he ANDL in IN environmen is vlided by boh simulion nd heoreicl resuls. The performnce of he ANDL in fixed SIR nd differen durion of IN versus Eb/N is shown in Fig. 1. As expeced,

7 7 Fig. 8: Comprison of differen pproches in ime nd frequency domin. /N = 1 db, SIR = db, λ = B s db 1-1 Oupu SNR ANDL (green solid lines) LIN (blue doed lines) BLN (red solid lines ) CLP (blck doed lines ) 5.3 db 7.4 db 7.8 db Impulsive noise o herml noise in bsebnd Fig. 9: Comprison of oupu SNR for differen pproches. λ = B s Simulion, s =.5 s Simulion, s =.5 s Simulion, s = 1 s Simulion, s = 1.5 s Theoreicl AWGN Theoreicl /N Fig. 1: versus /N. SIR = db, λ = B s. we hve beer performnce in shor durion IN. Fig. 11 shows he performnce of he ANDL in fixed durion of IN nd differen vlues of SIR versus Eb/N. As shown in Fig. 1 nd Fig. 11, he heoreicl resuls re well ligned wih simulion in differen scenrios which vlide our heoreicl clculions. Fig. 1 compres he performnce of ANDL wih blnking nd clipping for differen levels of impulsiviy (λ) wih τ s = 1µs. Fig. 1 shows h blnking nd clipping re very vulnerble o impulsiviy level nd heir performnce is drmiclly poor in high impulsive environmen. Alhough, he performnce loss of he ANDL wih incresing he impulsiviy level is lso noiceble, sill ouperforms oher pproches in ll scenrios. In Fig. 13, he performnce of ANDL for differen vlues of SIR in highly impulsive environmens (λ = B s ) is compred wih blnking nd clipping. Fig. 13 shows h boh blnking nd clipping hve poor performnce nd ANDL ouperforms hem especilly high SNR. The poency of ANDL in reducing he PSD of IN in he signl pssbnd is due o he fc h unlike oher nonliner mehods, ANDL is implemened in he nlog domin where he ouliers re sill brodbnd nd disinguishble. Therefore, in highly impulsive environmen s shown in Fig. 13, ANDL is highly preferble o digil pproches such s blnking nd clipping. VI. CONCLUSION In his work, n dpive nlog inermienly nonliner filer, referred o s Adpive Nonliner Differenil Limier

8 Simulion, SIR = 3 db Simulion, SIR = db Simulion, SIR = -3 db Theoreicl AWGN Theoreicl ANDL, SIR = 3 db ANDL, SIR = db ANDL, SIR = -3 db Theoreicl AWGN BLN, SIR = -3 db BLN, SIR = db BLN, SIR = 3 db CLP, SIR = -3 db CLP, SIR = db CLP, SIR = 3 db /N Fig. 11: versus /N. λ = B s, τ s = 1µs /N Fig. 13: comprison of ANDL, BLN, nd CLP versus /N for differen vlues of SIR. λ = B s, τ s = 1µs /N Fig. 1: comprison of ANDL, BLN, nd CLP versus /N for differen vlues of λ. SIR = db, τ s = 1µs. (ANDL) is proposed o miige impulsive noise (IN) in OFDM-bsed sysems. In ddiion, n pproximion of he ANDL using piecewise combinion of liner filers is used o derive closed-form nlyicl expressions for he verge signl-o-noise rio (SNR) he oupu of he proposed filer. We lso show h he heoreicl resuls re well ligned wih simulion resuls for differen composiions of noise. The heoreicl nlysis nd simulion resuls show h he ANDL ensures significn improvemen in SNR or performnce in he presence of srong IN componen. Moreover, he ANDL ouperforms oher convenionl oulier miigion mehods h exploi mpliude disribuion such s blnking nd clipping by providing higher oupu SNR nd lower in IN environmens. I is imporn o noe h he proposed ANDL is olly blind nd cn be deployed in rel-ime pplicions for boh sprse nd bursy IN scenrios. REFERENCES [1] J. A. C. Binghm, Mulicrrier modulion for d rnsmission: An ide whose ime hs come, IEEE Commun. Mg., vol. 8, no. 5, pp. 5 14, My [] HomePlug Green PHY The Sndrd For In-Home Smr Grid Powerline Communicions, Sd., 1. [3] IEEE Sndrd for Informion echnology-locl nd meropolin re neworks-specific requiremens-pr 11: Wireless LAN Medium Access Conrol (MAC) nd Physicl Lyer (PHY) Specificions Amendmen 6: Wireless Access in Vehiculr Environmens, Sd., Jul. 1. [4] X. Kui, H. Sun, S. Zhou, nd E. Cheng, Impulsive noise miigion in underwer cousic OFDM sysems, IEEE Trns. Veh. Technol., vol. 65, no. 1, pp , Oc. 16. [5] M. Ghosh, Anlysis of he effec of impulse noise on mulicrrier nd single crrier QAM sysems, IEEE Trns. Commun., vol. 44, no., pp , Feb [6] D. Umehr, H. Ymguchi, nd Y. Morihiro, Turbo decoding in impulsive noise environmen, in Globl Telecommun. Conf., vol. 1, Nov. 4, pp [7] H. Nkgw, D. Umehr, S. Denno, nd Y. Morihiro, A decoding for low densiy priy check codes over impulsive noise chnnels, in Proc. IEEE In. Symp. on Power Line Commun. nd is Appl., Apr. 5, pp [8] G. Ndo, P. Siohn, M. H. Hmon, nd J. Horrd, Opimizion of urbo decoding performnce in he presence of impulsive noise using sof limiion he receiver side, in Globl Telecommun. Conf., Nov. 8, pp [9] M. Nssr, J. Lin, Y. Morzvi, A. Dbk, I. H. Kim, nd B. L. Evns, Locl uiliy power line communicions in he 3 5 khz bnd: Chnnel impirmens, noise, nd sndrds, IEEE Signl Process. Mg., vol. 9, no. 5, pp , Sep. 1. [1] A. Al-Dweik, A. Hzmi, B. Shrif, nd C. Tsimenidis, Efficien inerleving echnique for OFDM sysem over impulsive noise chnnels, in Proc. IEEE In. Symp. on Personl, Indoor nd Mobile Rdio Commun., Sep. 1, pp [11] S. Liu, F. Yng, nd J. Song, An opiml inerleving scheme wih mximum ime-frequency diversiy for plc sysems, IEEE Trns. Power Del., vol. 31, no. 3, pp , June 16. [1] G. Cire, T. Y. Al-Nffouri, nd A. K. Nrynn, Impulse noise cncellion in OFDM: An pplicion of compressed sensing, in IEEE In. Symp. Inform. Theory., Jul. 8, pp [13] T. Y. Al-Nffouri, A. A. Qudeer, nd G. Cire, Impulse noise esimion nd removl for OFDM sysems, IEEE Trns. Commun., vol. 6, no. 3, pp , Mr. 14. [14] S. Liu, F. Yng, W. Ding, nd J. Song, Double kill: Compressivesensing-bsed nrrow-bnd inerference nd impulsive noise miigion for vehiculr communicions, IEEE Trns. Veh. Technol., vol. 65, no. 7, pp , Jul. 16. [15] S. Liu, F. Yng, X. Wng, J. Song, nd Z. Hn, Srucured-compressedsensing-bsed impulsive noise cncelion for mimo sysems, IEEE Trns. Veh. Technol., vol. 66, no. 8, pp , Aug 17. [16] J. Lin, M. Nssr, nd B. L. Evns, Impulsive noise miigion in powerline communicions using sprse Byesin Lerning, IEEE J. Sel. Ares Commun., vol. 31, no. 7, pp , Jul. 13.

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