Level crossing rate of relay system over LOS multipath fading channel with one and two clusters in the presence of co-channel interference

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1 INFOTEHJAHORINA Vol 5, March 6 Lvl crossng rat of rla sstm ovr LOS multath fadng channl th on and to clustrs n th rsnc of cochannl ntrfrnc Goran Ptovć, Zorca Nolć Facult of Elctronc Engnrng Unvrst of Nš Nš, Srba gorantovc969@gmalcom, orcanlc@lfanacrs Časlav Stfanovć, Danjl Đošć Facult of Natural Scncs and Mathmatcs Unvrst of Prštna Kosovsa Mtrovca, Srba caslavstfanovc@gmalcom, danjldjosc@gmalcom Dušan Stfanovć Collag of Ald tchncal Scncs Nš, Srba dusanstfanovc@tcntarrs Maro Carć Unvrst Busnss Acadm n Nov Sad Nov Sad, Srba carcmaro@ahoocom Abstract rlss rla mobl communcaton sstm th to sctons oratng ovr lnofsght (LOS) short trm fadng channl n th rsnc of cochannl ntrfrnc affctd to multath fadng s consdrd Dsrd sgnal and cochannl ntrfrnc roagat through nvronmnt th no mor than to clustrs Probablt dnst functon (PDF), cumulatv dstrbuton functon (CDF) and lvl crossng rat (LCR) of roosd sstm outut sgnal ar calculatd Drvd rssons can b usd for calculaton of outag robablt and avrag fad duraton of rla sstm Th nflunc of Rcan factor and ntrfrnc avrag or on lvl crossng rat ar anald and dscussd Kordslvl crossng rat (LCR); Rcan fadng; lnofsght (LOS); cochannl ntrfrnc I INTRODUCTION Small scal fadng and cochannl ntrfrnc ar rsntd n rlss communcaton sstm rsultng n outag robablt and avrag fad duraton dgradaton Rcan dstrbuton can b usd to dscrb sgnal nvlo n short trm fadng nvronmnt th lnofsght comonnt n th rsnc of on clustr Ths dstrbuton has Rcan factor hch can b valuatd as th rato of dsrd sgnal domnant comonnt or and dsrd sgnal scattrng comonnt or hn aramtr gos to ro, Rcan multath fadng channl rducs to Ralgh multath fadng channl, hn Rcan factor gos to nfnt Rcan multath fadng channl gos to no fadng channl [][] In ths ar Rcan dstrbuton s dnotd th hl dstrbuton hch can dscrb fadng channl th domnant comonnt and to clustrs s dnotd th Thr ar mor multath fadng channls hch can b dscrbd b usng and dstrbutons B sttng for, th fadng channl bcoms Naagamm channl th to clustrs In ths ar rlss rla mobl communcaton sstm th to sctons oratng ovr or short trm multath fadng channl n th rsnc of cochannl ntrfrnc also subjctd to or multath fadng s consdrd Undr dtrmnd condtons, sgnal nvlo at outut of rla sstm th to sctons can b rttn as roduct of sgnal nvlos of th frst and th scond scton In cllular mobl rlss sstm, th nflunc of cochannl ntrfrnc on sstm rformanc s sgnfcantl hghr comard to Gaussan nos, so th Gaussan nos ffcts can b nglctd In ntrfrnc lmtd nvronmnt, for th roosd rla modl, th rato of sgnal to ntrfrnc (SIR) can b calculatd as th rato of th roduct of to random varabls and on random varabl [5][8] In ths modl, random varabls hav or dstrbutons In ar [9], rlss rla communcaton sstm oratng ovr Ralgh short trm multath fadng channl s consdrd Lvl crossng rat of outut random rocss s calculatd and drvd ntgrals ar solvd b usng Lalac aromaton thorm Th rlss rla communcaton sstm n th rsnc of Naagamm multath fadng s studd n or [] In ths ar, sgnal at outut s calculatd as th roduct of sgnals at sctons In or [], statstcs of th rato of th roduct of to random varabls and random varabl ar valuatd In ar [], th frst ordr statstcal masurs b usng Mar functons ar ffcntl calculatd In ths ar, lvl crossng rat of outut sgnal of consdrd rla sstm n th rsnc of cochannl ntrfrnc s ffcnc calculatd Morovr, th jont robablt dnst functon of outut sgnal and th frst drvatv of outut sgnal s valuatd Obtand rssons can b usd for calculaton of avrag fad duraton of roosd rla sstm th to sctons To th bst of author s noldg, lvl crossng rat of mobl rla rlss sstm n th rsnc of or multath short

2 trm fadng and or cochannl ntrfrnc s not rortd n on tchncal ltratur II PDF OF OUTPUT SIR rlss rla sstm th to sctons n th rsnc of short trm multath fadng and cochannl ntrfrnc also subjctd to small scal fadng s consdrd Dsrd sgnal outut of rla sstm s rssd as roduct of to random varabls Th rato of dsrd sgnal and ntrfrnc can b modld as th rato of th roduct of to random varabls and and random varabl : Th condtonal PDF of s:, () Probablt dnst functon of s: æ ( ) ( ) ( ( ) æ ( ) ( ( ) æ ( ) è ( d d ( ) ( ) ( ) ò ò (7) d æ ( ) d d hr: d Aftr substtuton and avragng, th PDF of bcoms: æ d d () ò ò () Th random varabls (), () and () hav dnst robablt: ( ) ( ) æ I ( ) æ ( ) ( ( ) ( ) æ( ) è ( ( ) æ ( ) è ( () (5) (6) Th tofold ntgral s: J òd òd æ ( ) ( ) ò d ( ) ( ) (( ) ( ) ) ( ) ò d (( ) ( ) ) ( ) ( ) s, ( ) ( ) ( ) s, d ds ( ) ( ) Aftr substtutng th ntgral J bcoms: ( ) æ( ) ( ) ( ) J ( ) dss ( s) ( ) ( ) ( ) s (8) (( ) ) ò (9) Prvousl ntgral can b solvd b usng formula []: 5

3 dtt a ct (,, ) a b ò G t au abc () ( ) æ( ) J (( ) ) ( ) è ( ) () æ ( )( ) G ( ) U,, ( ) III Th SIR at outut of rlss rla communcaton sstm s: LEVEL CROSSING RATE OF OUTPUT SIR, () Th frst drvatv of s: & & & & () Th frst drvatv of dstrbuton has Gaussan dstrbuton Also, lnar transformaton of Gaussan random varabls has Gaussan dstrbuton Th man of &s ro Th varanc of &s: hn: s & s & s & s & (),, s & fm s & fm s & fm (5) d hr: d B substtutng th & & s: & æ d d (9) ( ) ( ) ( ) & ò ò & & LCR of &s: æ N d d d ò && & & ò ò fm 8( )( )( ) s & æ ( ) æ ( ) è ( è æ ( d d è ò ò ( ( ) ( ) ( ) ( ) ( ) ( ) () Prvousl tofold ntgral can b solvd b usng Lalac aromaton formula []: l f (, ) g (, ) l f (, ) d dg(, ) l b(, ) ò ò () hr and soluton of quatons: Aftr substtuton th rsson for varanc of &bcoms: æ s fm & (6) Th jont PDF of & and s: ( & ) ( & ) ( ) (7) & & and: f, f,, f f (, ) (, ) b f(, ) f(, ) () () hr: d æ d (8) IV LEVEL CROSSING RATE OF RELAY SYSTEM OVER K MULTIPATH FADING OUTPUT SIR Th random varabls, and follo κµ dstrbuton: 6

4 ( ) ( ) ( ) ( ) I æ, ³! G ( ) æ nflunc of Rcan factor on lvl crossng rat s mor domnant for hghr valus of Rcan factor Morovr, th nflunc of Rcan factor on lvl crossng rat s hghr for hghr valus of sgnal to ntrfrnc rato hn SIR ncrass, avrag lvl crossng rat ncrass, rachs ts mamum, than avrag lvl crossng rat dcrass for hghr valus of SIR hn Rcan factor ncrass, mamum of curv, slghtl gos to hghr valus of sgnal to ntrfrnc rato ( ) ( ) æ ( )! G ( ) ( ) æ ( ) ( )! G ( ) () Lvl crossng rat (LCR) of outut SIR s: fm ( ) æ ( ) N G ( ) æ ( )! ( ) G æ ( ) ( )! G ( ) òdòd ( ) ( ) ( )! (5) ( 7)ln ( )ln Prvousl tofold ntgral s solvd b usng Lalac aromaton formula [] Fgur Normald lvl crossng rat (LCR), hn Ω Ω Ω and dffrnt valus of In fgur, lvl crossng rat of rlss sstm vrsus sgnal to ntrfrnc rato s shon, for th cas hn Rcan factor s constant and sgnal nvlos avrag ors hav dffrnt valus V NUMERICAL RESULTS In fgur, avrag lvl crossng rat of rlss rla mobl communcaton sstm th to sctons oratng ovr multath fadng channl n th rsnc of cochannl ntrfrnc vrsus sgnal to ntrfrnc (SIR) s shon Consdrd s cas hn Rcan factor valus ar,, and 6 and sgnal nvlos avrag ors ar constant Lvl crossng rat ncrass as Rcan factor ncrass Th Fgur Normald lvl crossng rat (LCR) dndng on th nvlo, hn and dffrnt valus Ω, Ω and Ω Smlarl, lvl crossng rat ncrass for lor valus of sgnal to ntrfrnc rato and lvl crossng rat dcrass for 7

5 hghr valus of sgnal to ntrfrnc rato hn or of sgnal nvlos n both sctons ncras, lvl crossng rat of rla sstm ncrass for hghr valus of sgnal to ntrfrnc rato hl lvl crossng rat dcrass for lor valus of sgnal to ntrfrnc rato hn or of ntrfrnc ncrass, avrag lvl crossng rat ncrass for lor valus of SIR hl lvl crossng rat dcrass for hghr valus of SIR Morovr, hn or of ntrfrnc ncrass mamum gos to lor valus of SIR In fgur, lvl crossng rat vrsus sgnal to ntrfrnc rato s rsntd for th cas hn rlss rla sstm oratng ovr multath fadng channl n th rsnc of cochannl ntrfrnc Fgur Normald lvl crossng rat (LCR) dndng on th nvlo, for dffrnt valus, Ω, Ω and Ω CONCLUSION In ths ar rlss rla mobl sstm th to sctons oratng ovr lnofsght multath fadng channl th on or to clustrs n th rsnc of cochannl ntrfrnc affctd to multath fadng s consdrd Sgnal roagats n th channl th on clustr dscrbd th dstrbuton or to clustrs dscrbd b usng dstrbuton For consdrd modl, sgnal to ntrfrnc rato s rsntd as rato of roduct of to random varabls and random varabl Th dstrbuton dscrb sgnal nvlo n channl th lnofsght comonnts and on clustr and dstrbuton fadng channl th to clustrs Closd form rssons for PDF and CDF of rato of roduct of to or random varabls and or random varabl ar calculatd Thos rsults can b usd for valuaton of outag robablt and bt rror robablt of roosd rla sstm Furthrmor, avrag lvl crossng rat of rla sstm th to sctons n th rsnc of cochannl ntrfrnc s calculatd Ths rsson for LCR can b usd for calculaton of avrag fad duraton of consdrd rla sstm B sttng for n rssons for PDF, CDF and LCR can b drvd rssons for PDF, CDF and LCR of rla sstm th to scton oratng ovr Ralgh or Naagamm short trm fadng channls n th rsnc cochannl ntrfrnc subjctd to Ralgh or Naagamm short trm fadng Th nflunc of Rcan factor and avrag valu of sgnal nvlo at lvl crossng rat s analsd and dscussd Lvl crossng rat dcrass hn Rcan factor ncrass REFERENCES [] S Panc, Fadng and Intrfrnc Mtgaton n rlss Communcatons, USA, CRC Prss, [] M K Smon, M S Aloun, Dgtal Communcaton ovr Fadng Channls, USA: John l & Sons [] Dong X Baulu Avrag lvl crossng rat and avrag fad duraton of slcton dvrst, IEEE Communcaton Lttrs; (5); 969, [] Djan Mlc, Danjl Djosc, Caslav Stfanovc, Stfan Panc and Mhajlo Stfanovc, Scond ordr statstcs of th SC rcvr ovr Rcan fadng channls n th rsnc of multl Naagamm ntrfrrs, Intrnatonal Journal of Numrcal Modllng, Onln ISSN: 99, ISSN: 897, DOI: /jnm65, Ar 5 [5] A Panajotovc, N Sulovc, M Stfanovc, D Draca, Avrag Lvl Crossng Rat of Dual Slcton Dvrst ovr Corrlatd Unbalancd Naagamm Fadng Channls n th Prsnc of Cochannl Intrfrnc, IEEE Communcatons Lttrs, vol 6 br 5, str 6969 [6] Srboljub Zdravovc, Danjla Alsc, Ivca Marjanovc, Goran Ptovc, Danjl Djosc and Prdrag Mlacc, Lvl crossng rat of rato of roduct to m random varabls and m random varabl valuatd b Lalac aromaton formula, th Intrnatonal Confrnc on Ald Elctromagntcs ПЕС 5 August Stmbr, 5, Nš, Srba, ISBN [7] Goran Ptovc, Danjl Djosc, Vladta Mlnovc, Dragan Radnovc, Ivana Dnc, Lvl crossng rata of rato of roduct to Naagamm random varabls and Naagamm random varabl, UNITECHGABROVO 5, Vol, ags 776, Novmbr 5 ISSN: X [8] Goran Ptovć, Stfan Panć, Branmr Jašć, Lvl crossng rat of macrodvrst th thr mcrodvrst SC rcvrs ovr Gamma shadod Naagamm channl, Unvrst Thought Publcaton n Natural Scncs, vol 6, No, 5, do:597/unvtho69797 [9] Z HadVlov, N Zlatanov, G K Karagannds, On th scond ordr statstcs of th multho Ralgh fadng channl, acctd for ublcaton n th IEEE Transactons on Communcatons, Volum 57, Issu 6, Jun 9 Pag(s):85 8, DOI: 9/TCOMM9676 [] Nola Zlatanov, Zoran HadVlov, Gorg K Karagannds, Lvl Crossng Rat and Avrag Fad Duraton of th Doubl Naagamm Random Procss and Alcaton n MIMO Khol Fadng Channls, IEEE Communcatons Lttrs, Volum, Issu, Pag(s):8 8, Novmbr 8, DOI: 9/LCOMM8858 [] E Mc, N Sulovc, M Bandjur, M Stfanovc, P Salvc, Th Dstrbuton of Rato of Random Varabl and Product of to Random Varabls and ts Alcaton n Prformanc Analss of MultHo Rlang Communcatons ovr Fadng Channls, Prglad Eltrotchncn, vol 88, no 7A, 7, [] A Matovc, E Mc, N Sulovc, M Stfanov, M Matovc, C Stfanovc, Th dstrbuton of th rato of th roducts of to ndndnt αµ varats and ts alcaton n th rformanc analss of rlang comuncaton sstm, Mathmatcal Problms n Engnrng, ISSN: 5657, Hnda Publshng, vol, Artcl ID 76, [] I Gradshtn, I, Rh Tabls of Intgrals, Srs, and roducts Acadmc Prss, N Yor 99 8

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