The Second Order Performance of Macrodiversity Reception in the Presence of Weibull Fading, Gamma Fading and α-κ-µ Co-channel Interference

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1 Dragana Krstć t al. Intrnatonal Journal of Communcatons Th Scond Ordr Prformanc of Macrodvrst Rcpton n th Prsnc of Wbull Fadng, Gamma Fadng and α-κ-µ Co-channl Intrfrnc DRAGANA KRSTIĆ, SINIŠA MINIĆ*, SUAD SULJOVIĆ, MIHAJLO STEFANOVIĆ Facult of Elctronc Engnrng, Unvrst of Nš Alsandra Mdvdva 4, 8 Nš *Unvrst of Prstna, Kosovsa Mtrovca Tachrs' tranng facult n Prrn-Lposavć Nmanjna bb, 388 Lposavc SERBIA dragana.rstc@lfa.n.ac.rs Abstract: - In ths papr, th wrlss sstm consstng of macrodvrst slcton combnng (SC) rcvr and two mcrodvrst SC rcvrs undr th nflunc of small scal fadng and larg scal fadng, as wll as co-channl ntrfrnc s obsrvd. Small scal fadng has Wbull dstrbuton. Corrlatd larg scal fadng s dscrbd b Gamma dstrbuton. Co-channl ntrfrnc s dsturbd b α-κ-µ fadng and Gamma larg scal fadng. Probablt dnst functon (PDF) and cumulatv dstrbuton functon (CDF) of th rato of Wbull random varabl and α-κ-µ random varabl ar gvn. Th formula for CDF of macrodvrst SC rcvr output sgnal to ntrfrnc rato (SIR) s also prsntd. Lvl crossng rats at th outputs of mcrodvrst SC rcvrs ar dtrmnd. Thn, th lvl crossng rat (LCR) of wrlss sstm output sgnal to ntrfrnc rato s drvd and shown n som fgurs. Basd on thm, th nflunc of Wbull fadng nonlnart paramtr, α-κ-µ fadng svrt paramtr, α-κ-µ fadng nonlnart paramtr, Rcan factor, Gamma long trm fadng svrt paramtr and Gamma long trm fadng corrlaton coffcnt s studd. K-Words: - macrodvrst rcvr; mcrodvrst rcvr; slcton combnng (SC); Gamma fadng; Wbull fadng; α-κ-µ fadng; lvl crossng rat Introducton Th charactrstcs of th frst and th scond ordrs of th wrlss sstm ar mpard owng to th nflunc of small scal fadng, larg scal fadng fadng and co-channl ntrfrnc. Th scond ordr prformancs ar lvl crossng rat (LCR) and avrag fad duraton (AFD) of wrlss mobl communcaton sstm []. To mtgat th small scal fadng, larg scal fadng fadng and cochannl ntrfrnc ffcts on th lvl crossng rat t s ncssar to us a macrodvrst sstm. Th macrodvrst sstm conssts of on macrodvrst rcvr and two or mor mcrodvrst rcvrs []. Th macrodvrst rcvr s th most oftn of slcton combnng (SC) tp. Its SC rcvr slcts mcrodvrst rcvr wth hghr sgnal nvlop (or sgnal to ntrfrnc rato) avrag powr at nputs, rsultng n Gamma larg scal fadng rducton. Th mcrodvrst rcvrs could b mamal rato combnng (MRC), qual gan combnng (EGC), or slcton combnng rcvrs [3]. Th smplst s SC rcvr whch chooss th branch wth th hghst sgnal, or sgnal to ntrfrnc rato, whch mpls th small scal fadng ffcts rducton and co-channl ntrfrnc ffcts rducton []. Man dffrnt dstrbutons dscrb sgnal nvlop n fadng nvronmnts such as Ralgh, Rcan, Naagam, Wbull dstrbuton, or gnral dstrbutons l α-κ-µ dstrbuton [4]. Wbull dstrbuton dscrbs small scal sgnal nvlop varaton n nonlnar, non ln of sght (NLoS) fadng surroundngs [5]. It has paramtr α, calld nonlnart paramtr. Wbull dstrbuton s gnral dstrbuton bcaus for α=, Wbull dstrbuton rducs to Ralgh dstrbuton. For α tnds to nfnt Wbull channl bcoms no fadng channl. On th othr sd, th α-κ-µ dstrbuton s charactrd b thr paramtrs, whr paramtr α s nonlnart paramtr, κ s Rcan factor and µ s small scal fadng svrt paramtr [6]-[9]. Rcan factor s th rato of domnant componnt powr and scattrng componnts powrs. ISSN: Volum, 7

2 Dragana Krstć t al. Intrnatonal Journal of Communcatons Th α-κ-µ dstrbuton s also gnral dstrbuton. Th othr fw dstrbutons could b obtand from α-κ-µ dstrbuton as a spcal cas. If α=, ths dstrbuton rducs to κ-µ dstrbuton; f κ=, th α-κ-µ bcoms α-µ dstrbuton; f κ= and µ=, th α-κ-µ dstrbuton coms down to Wbull dstrbuton; f α= and µ=, th α-κ-µ dstrbuton rducs to Rcan dstrbuton; f α= and κ=, th α- κ-µ s Naagam-m dstrbuton, and f α=, κ= and µ=, th Ralgh dstrbuton occurs. Th analss of wrlss sstms n α--μ fadng nvronmnt subjctd to shadow ffct s prsntd n [8]. Th scond ordr prformanc of th α-κ-μ nvlop s drvd n [9]. Man artcls gvn n ltratur rvw consdr macrodvrst sstm prformanc n th prsnc of larg scal fadng, small scal fadng and co-channl ntrfrnc. For ampl, th wrlss sstm wth macrodvrst SC rcvr and two mcrodvrst SC rcvrs n Gamma shadowd Wbull multpath fadng nvronmnt, undr Wbull co-channl ntrfrnc nflunc, s consdrd n []. Th lvl crossng rat s calculatd. Macrodvrst sstm wth macrodvrst SC rcvr and two mcrodvrst SC rcvrs whr dsrd sgnal s undr Wbull short trm fadng, corrlatd Gamma long trm fadng and co-channl ntrfrnc, whch prncs α-κ-µ short trm fadng and corrlatd Gamma long trm fadng s anald n []. Hr, th outag probablt s valuatd from cumulatv dstrbuton functon. In ths artcl, th wrlss sstm wth macrodvrst SC rcvr, usd to mtgat fadng and co-channl ntrfrnc ffcts on th sstm prformanc of th scond ordr, s anald. Usful sgnal suffrs Wbull fadng and Gamma fadng, and co-channl ntrfrnc s undr α-κ-µ fadng and Gamma larg scal fadng. Lvl crossng rat of th rato of Wbull random varabl and α-κ-µ random varabl s drvd and usd for calculaton th LCR of mcrodvrst and macrodvrst SC rcvrs. PDF of th Rato of Wbull Random Varabl and α-κ-µ Random Varabl Th rato of Wbull random varabl and α-κ-µ random varabl s:,,, () whr follows Ralgh dstrbuton: p, ; () Ω s avrag powr of. follows Wbull dstrbuton [5]: p, ; (3) α s Wbull short trm fadng nonlnart paramtr. Th random varabl follows κ-µ dstrbuton [] [3]: p,!, (4) and random varabl has α-κ-µ dstrbuton [6]: p!, (5) whr κ s Rcan factor; α s nonlnart paramtr; Ω s avrag powr of ; Γ(.) s ncomplt gamma functon [4]. Th frst drvatv of s: / (6) Th frst drvatv of Ralgh random varabl has Gaussan dstrbuton and th frst drvatv of κ-µ random varabl has also Gaussan dstrbuton. Thrfor, and ar Gaussan varabls. Snc th lnar transformaton of Gaussan random ISSN: Volum, 7

3 Dragana Krstć t al. Intrnatonal Journal of Communcatons varabls has also Gaussan dstrbuton, follows Gaussan dstrbuton. Th man of s: bcaus. Th varanc of s: whr / (7) f m (8) (9) f m () Aftr substtutng, th prsson for varanc of bcoms: 4 fm. () Th jont probablt dnst functon of, and s: p p / p wth: / / p p p () Th p / s: p p d / d (3) d. (4) d Aftr th nt substtutng, th prsson for p s: / p p p p / (5) Th jont probablt dnst functon of and s: p d p / / dp p p (6) Lvl crossng rat of random procss s: N d p / / d p p d p / d p p / d p p f m / / / / /,! f d m / / / /! / /. (7) 3 Lvl Crossng Rat of Macrodvrst Sstm Output SIR Th modl of macrodvrst sstm obsrvd n ths papr s prsntd n Fg.. Probablt dnst functon of j, =,; j=, s Wbull (3). j j p,,,,, j j j j. ISSN: Volum, 7

4 Dragana Krstć t al. Intrnatonal Journal of Communcatons SC SC,,,, SC,, whr c s Gamma long trm fadng svrt paramtr of ntrfrnc, β s avrag valu of s and s. Th CDF of macrodvrst SC rcvr output sgnal to ntrfrnc rato s [, q. (3)]: /, F d s p s d F s p d s d s ps s d d F /, s p Fg.. Sstm modl Random varabls j follows α-κ-µ dstrbuton (5): p j j s! j s,.` j Jont probablt dnst functon (JPDF) of Ω and Ω s []: p c c c c 3 3! 3 3 c 3c 3c,, (8) whr c s Gamma long trm fadng svrt paramtr, ρ s corrlaton coffcnt, and Ω s avrag valu of Ω and Ω. Th jont probablt dnst functon (JPDF) of s and s s []: pss s s s c j s c c s c s, s, s c c (9) s d s p s d F /, s p d 3 cj 3 c j 3 c j 3 c j c j 3 c j 3 F 3 c j,,3 c j ; 3 c j 3 c j c j 3 c j 3 F 3 c j,,3 c j ; 3 c j c j 3 c j 3 c j 3 ISSN: Volum, 7

5 Dragana Krstć t al. Intrnatonal Journal of Communcatons F 3 c j,,3 c j ; () Macrodvrst sstm chooss mcrodvrst SC rcvr wth hghr sgnal nvlop avrag powr or sgnal to ntrfrnc rato from ts nputs. Th lvl crossng rat at th SC rcvr output s: N F N! / / fm /! / () Th lvl crossng rat of macrodvrst SC rcvr output sgnal to ntrfrnc rato procss s: / N ds p s d d N s p s / ds p s d d N s p s ds p s d d N / s p s c c c c c c 3 3! 3 3 c! / f m / /! c / ds s 3 c / d s s 3 c d c c c c c c 3 3! 3 3 c! s / f m / /! j 3 c 3 c j d s c / ds s 3 c / 3 c j j ISSN: Volum, 7

6 Dragana Krstć t al. Intrnatonal Journal of Communcatons s s () In th prvous prsson, two-fold ntgral can b wrttn n th form [5] [6]: d J ds s s c / 3 c / 3 c j s s Th ntgral J s: J J s c / J ds s c / j d 3 s (3) (4) Th ntgral can b solvd b usng th formula: p s p dss d a b p n p p p p n abs n p p n p p p a F p p n, p ; p p ; b (5) Hr, F (α,β;γ;) s a Gauss's hprgomtrc functon (Gauss 8, Barns 98). In gnral, t arss th most frquntl from phscal problms drvd b Gauss and Barns n [7] and [8]:!! 3! 3... Ths s so-calld rgular soluton, dnotd b: F, ; ; (6)...!! n n n n. (7) n! It convrgs f c s not a ngatv ntgr for all of and s on th unt crcl f Rc a b. Hr, a n s a Pochhammr smbol. For soluton of th ntgral J, th paramtrs ar: p c / n p c / j 3 a b n p n c 3 3 / p p 3c j 3 p p n 3c j 3 Aftr ntroducng ths paramtrs n (5), th ntgral J bcoms: J 3 3 / c c / c / j 3 3c j 3 3c 3 j c / 3c j 3 F 3c 3 j, c / ;3c 3 j; Th ntgral J s: s c / J ds s (8) ISSN: Volum, 7

7 Dragana Krstć t al. Intrnatonal Journal of Communcatons c 3 / j d s (9) It could b solvd n th sam wa, wth th nt group of paramtrs whch wll b put nto (5): p c / p c / j 3 a b n p n c 4 3 / p p 3c j 3 p p n 3c j 3 Th ntgral J s: J 4 3 / c c / c / j 3 3c 3 j 3c 3 j c / 3c j 3 F 3c 3 j, c / ;3c 3 j; (3) Fnall, aftr substtuton J from (8) and J from (3) nto (3) and (), th lvl crossng rat of macrodvrst SC rcvr output sgnal to ntrfrnc rato bcoms: N c c c c c c 3 3! 3 3 c! / f m / /! j 3 c 3 c j 3 3 / c c / c / j 3 j 3c j 3 3c 3 j c / 3c j 3 F 3c 3 j, c / ;3c 3 j; 4 3 / c c / c / j 3 3c 3 j 3c 3 j c / 3c j 3 F 3c 3 j, c / ;3c 3 j; (3) 4 Numrcal rsults Fg. to 4. prsnt avrag lvl crossng rat of macrodvrst sstm for svral valus of Gamma long trm fadng svrt paramtr c, Wbull short trm fadng nonlnart paramtr α, α-κ-µ short trm fadng nonlnart paramtr α, α-κ-µ short trm fadng Rcan factor κ and α-κ-µ short trm fadng svrt paramtr µ. In Fg., th avrag LCR s plottd vrsus output sgnal to ntrfrnc rato. ISSN: Volum, 7

8 Dragana Krstć t al. Intrnatonal Journal of Communcatons Fg.. Lvl crossng rat of macrodvrst sstm vrsus output sgnal to ntrfrnc rato Th avrag LCR ncrass for small valus of SC combnr output sgnal to ntrfrnc rato, achvs th mamum, than dclns. Th nflunc of SC combnr output sgnal to ntrfrnc rato on avrag LCR s bggr for hghr valus of Wbull multpath fadng svrt paramtrs. Avrag LCR has lowr valus for bggr valus of Wbull multpath fadng svrt paramtr α and thn sstm prformanc s bttr. Avrag LCR vrsus output SIR s shown n Fg. 3 for varabl avrag powr of dsrd sgnal. Th othr paramtrs ar: µ=c=β=, β =κ=α=. On can notc from ths fgur that whn avrag powr Ω ncrass, LCR ncrass too. Ths ncras s mor pronouncd for hghr valus of. Also, avrag LCR of macrodvrst SC rcvr output sgnal vrsus th SC rcvr output sgnal to ntrfrnc rato s prsntd n Fg. 4. Fg. 3. LCR of macrodvrst sstm vrsus output SIR Fg. 4. LCR of macrodvrst sstm vrsus SIR Hr, avrag powr of Gamma fadng s varabl and othr paramtrs ar constant. LCR has lowr valus for gratr valus of avrag powr of Gamma varabl β and SC combnr output sgnal to ntrfrnc rato. Sstm prformancs ar bttr for lowr valus of th avrag LCR. Ths fgurs for th lvl crossng rat vrsus sgnal nvlop ar drown to show th mpact of fadng paramtrs to th LCR and for choosng optmal paramtrs. 5 Concluson In ths papr, macrodvrst sstm wth macrodvrst SC rcvr and two mcrodvrst SC rcvrs n th prsnc of small scal fadng and larg scal fadng s obsrvd. Dsrd sgnal s hndrd b Wbull multpath fadng and corrlatd Gamma larg scal fadng, and ntrfrnc s undr th affct of α-κ-µ small scal fadng and Gamma larg scal fadng. Macrodvrst sstm mtgats all of thm, small and larg scal fadng ffcts, and co-channl ntrfrnc ffcts on th lvl crossng rat. Macrodvrst SC rcvr wors so that chooss mcrodvrst wth hghr sgnal to ntrfrnc rato. Mcrodvrst rcvrs combn SIRs from multpl antnnas from bas statons and slct th branch wth hghr SIR, and rducs small trm fadng and co-channl ntrfrnc. In th artcl, PDF and CDF of th rato of Wbull and α-κ-µ random varabls ar drvd. Thn, th prssons for cumulatv dstrbuton functons of SIR at outputs of mcrodvrst SC rcvrs and macrodvrst SC rcvr output SIR ar gvn, and fnall, th LCR of macrodvrst sstm n th prsnc of Wbull small scal fadng, Gamma larg scal fadng and α-κ-µ co- ISSN: Volum, 7

9 Dragana Krstć t al. Intrnatonal Journal of Communcatons channl ntrfrnc s prformd. Th nflunc of Wbull small scal fadng nonlnart paramtr, Gamma larg scal fadng svrt paramtr, α-κ-µ small scal fadng Rcan factor, α-κ-µ small scal fadng nonlnart paramtr and α-κ-µ small scal fadng svrt paramtr on LCR s anald. LCR dcrass whn Gamma long trm fadng svrt paramtr ncrass, Wbull short trm fadng svrt paramtr and α-κ-µ short trm fadng svrt paramtrs ncras and avrag powr dcrass. Acnowldgmnt Th papr s mad n th fram of projcts III-446 and TR-3335 of th Srban Mnstr of Educaton, Scnc and Tchnologcal Dvlopmnt. Rfrncs: [] S. Panc, M. Stfanovc, J. Anastasov, P. Spalvc, Fadng and Intrfrnc Mtgaton n Wrlss Communcatons. CRC Prss, USA, 3. [] M. K. Smon, M. S. Aloun, Dgtal Communcaton ovr Fadng Channls, USA: John Wl & Sons.. [3] W.C.Y. L, Mobl communcatons ngnrng, Mc-Graw-Hll, NwYor, USA, 3. [4] G. Fradnrach, M. D. Yacoub, Th α-η-μ and α-κ-μ Fadng dstrbutons, n Proc. of th 6 IEEE Nnth Intrnatonal Smposum on Sprad Spctrum Tchnqus and Applcatons (ISSSTA), Aug. 6, pp. 6-. [5] W. Wbull, A statstcal dstrbuton functon of wd applcablt, J. Appl. Mch.-Trans. ASME, 95, 8 (3): [6] P. C. Sofotasos, S. Frar, Th α-κ-μ/ gamma dstrbuton: A gnrald non-lnar multpath/shadowng fadng modl, IEEE Annual Inda Confrnc (INDICON), 6-8 Dc., Hdrabad, ISBN: , DOI:.9/INDCON , pp. -6. [7] P. C. Sofotasos, S. Frar, Th α-κ-μ Etrm dstrbuton: Charactrng non- lnar svr fadng condtons, Proc. of Australasan Tlcommuncaton Ntwors and Applcatons Confrnc (ATNAC),, Mlbourn, Australa, Nov., pp. -4. [8] S. Panc, P. Spalvc, A. Marovc, M. Stfanovc, Prformanc analss of wrlss communcaton sstm n α--μ nvronmnt subjctd to shadowng, Intrnatonal Confrnc «Mathmatcal and Informatonal Tchnologs, MIT-3», (X Confrnc «Computatonal and Informatonal Tchnologs for Scnc, Engnrng and Educaton»), Vrnjaca Banja, Srba, Sptmbr, 5-8, 3, Budva, Montngro, Sptmbr, 9-4, 3. [9] A. K. Papaafropoulos, S. A. Kotsopoulos, Scond-Ordr Statstcs for th Envlop of α- κ-μ Fadng Channls, IEEE Communcatons Lttrs, Vol. 4, Issu: 4, Aprl, pp [] D. Alsć, D. Krstć, Z. Popovć, M. Stfanovć, Lvl Crossng Rat of Macrodvrst SC Rcvr Output Procss n th Prsnc of Wbull Short Trm Fadng, Gamma Long Trm Fadng and Wbull Cochanll Intrfrnc, WSEAS Transactons on Communcatons, ISSN / E-ISSN: 9-74 / 4-864, Volum 5, 6, Art. #3, pp [] D. Krstc, S. Mnć, S. Mlosavljvć, B. Mlosavljvć, M. Stfanovć, Macrodvrst Outag Prformanc n th Prsnc of Wbull Short Trm Fadng, Gamma Long Trm Fadng and α-κ-µ Co-channl Intrfrnc, Intrnatonal Journal of Communcatons, Vol., 7, pp. 4-. [] M. D. Yacoub, Th κ-μ dstrbuton and th η- μ dstrbuton, IEEE Antnnas and Propagaton Magan, Volum: 49 Issu:, Jun 7, pp: 68 8, DOI:.9/MAP [3] U. S. Das, M. D. Yacoub, Th κ-μ phasnvlop jont dstrbuton, IEEE Transactons on Communcatons,, Vol. 58, Issu:, pp. 4 45, DOI:.9/TCOMM [4] M. Abramowt, I. A. Stgun, Handboo of Mathmatcal Functons wth Formulas, Graphs, and Mathmatcal Tabls, Appld Mathmatcs Srs, ds. (983) [Jun 964]. Chaptr 6.5, 55 (Nnth rprnt wth addtonal corrctons of tnth orgnal prntng wth corrctons (Dcmbr 97); frst d.). Washngton D.C.; Nw Yor: Untd Stats Dpartmnt of Commrc, Natonal Burau of Standards; Dovr Publcatons, ISBN LCCN , MR 6764, LCCN 65-53, "Incomplt Gamma functon", 6.5. [5] I. S. Gradshtn, I. M. Rh, Tabl of Intgrals, Srs and Products. Acadmc Prss, USA San Dgo,. [6] A. P. Prudnov, Y. A. Brchov, O. I. Marchv, Intgrals and Srs, Volum 3: ISSN: Volum, 7

10 Dragana Krstć t al. Intrnatonal Journal of Communcatons Mor Spcal Functons. st d., Gordon and Brach Scnc Publshrs, Nw Yor, 986. [7] C. F. Gauss, Dsqustons gnrals crca srm nfntam, Soctas Rga Scntarum Gottngnss, Burnd Lbrar [8] E. W. Barns, A Nw Dvlopmnt n th Thor of th Hprgomtrc Functons, Proc. London Math. Soc. 6, 4-77, 98. ISSN: Volum, 7