DUET Journal Vol., Issu, Jun Ergodc apac of a SIO Ssm Ovr Nakagam-q Fadng hannl d. Sohdul Islam * and ohammad akbul Islam Dp. of Elcrcal and Elcronc Engnrng, Islamc Unvrs of Tchnolog (IUT, Gazpur, Bangladsh * Emal : msohdul@u-dhaka.du ABSTAT In hs papr, w prsn an analcal xprsson for rgodc capac of sngl-npu mulpl-oupu (SIO wrlss communcaon ssms ovr Nakagam-q fadng channls. W drv analcal xprssons boh for dncall ndpndn dsrbud (..d. and non- dncall ndpndn dsrbud subchannls (non-..d. Th capac calculaon has bn carrd ou usng approxmaons on zroh-ordr modfd Bssl funcons of h frs knd. W prsn smulaon and numrcal rsul basd on small lm argumn approxmaon. W assumd ha channl sa nformaon (SI s avalabl a h rcvr, bu no a h ransmr. W found ha h rgodc capac of h Nakagam-q fadng subchannls s proporonal o h ransmd sgnal o nos rao of h subchannls.. INTODUTION In ral lf wrlss communcaon ssms, w xpc o download and upload nformaon fasr as possbl. To mak fasr w nd o ncras h capac. Informaon hor has dmonsrad h normous ponal capac of wrlss communcaon ssms wh mulpl annnas a boh ransmr and h rcvr, whn h channl xhbs rch scarng and channl-sa nformaon (SI s avalabl a h rcvr []. Th concp of nformaon horc capac was frs nroducd b Shannon [] o characrz fundamnal lms of communcaon ovr wrlss channl. In [3] h nvsgad nformaon ouag probabl of orhogonal spac-m block codd mulplnpu mulpl-oupu (IO ssms oprang ovr Ho dsrbud fadng channls for ndpndn and dncall dsrbud fadng channls. Th also nvsgad xacclosd form xprssons for non- ndpndn and dncall dsrbud fadng channls usng saddl pon approxmaon mhod o approxma nformaon ouag probabl. Hr n hs papr w prsn channl capac formulas of snglnpu mulpl-oupu (SIO ssm ovr Nakagam-q fadng channls. ajor of h rsarch rlad o SIO capac has focusd on algh, can and Nakagam-m dsrbuons. Thr ar ohr ps of fadng dsrbuons whch srvs good modls undr cran crcumsancs. I allows h modlng [] of propagaon nvronmn whou a domnan componn ovr h scard wavs (a suaon of Non-ln of Sgh. I has found applcaon n h rror prformanc valuaon of dgal communcaon ssms ovr gnralzd fadng channl [5]. Also, hs fadng channl s normall obsrvd n sall lnk subjc o srong onosphrc scnllaon and havl shadowd nvronmn [6]. Insananous sgnal o nos rao follows h log-squar-ho dsrbuon n nr-sall communcaons [7]. cnl, h Nakagam-q modl s 9 bng usd mor frqunl n prformanc analss and ohr suds rlad o mobl rado communcaons. Th rs of h paprs organzd as follows. Scon dscrbs h Nakagam-Ho dsrbuon. Ssm modl s prsnd n Scon 3. Scon shows h probabl dsrbuon funcon (PDF calculaons for boh nondncall ndpndn and dncall ndpndn Nakagam-q fadng channls. Th rgodc capac calculaons ar dscrbd n Scon 5. In Scon 6 smulaon and rsuls ar shown. Fnall, Scon 7 dscrbs h concludng rmarks of hs work.. NAKAGAI-q DISTIBUTION Th Nakagam-q dsrbuon s h dsrbuon of modulus of complx Gaussan random varabls wh zro man and unqual varancs. In [8], h probabl dsrbuon funcon (PDF of Nakagam-q dsrbuon s gvn b, ( q q ( q ( p ( I q ( Whr I (. s h zroh-ordr modfd Bssl funcon of h frs knd and q s h Nakagm-q fadng paramr rangs from o. and ar avrag SN and nsananous SN rspcvl. Th Nakagam-q dsrbuon spans h rang from on-sdd Gaussan fadng (q= o algh fadng whn q=. 3. SYSTE ODEL In sngl-npu mulpl-oupu (SIO wrlss communcaon ssm, ransmr s quppd wh sngl ransmng annna and rcvr wh mulpl rcvng annnas ( n. L P bng ransmd sgnal powr. L us consdr h rcvd sgnal s corrupd b addv wh
DUET Journal Vol., Issu, Jun gaussan nos (AWGN as wll as h ransmsson s carrd ovr non-dncall ndpndn Nakagam-q fadng subchannls. Th rcvd sgnal [9] vcor can b xprssd as r hx z ( whr h s a n vcor dfnng h subchannl gans from ransmr o ach rcvr annna, h h, h,..., h n, x s h ransmd sgnal, and z s an n vcor of an..d. zro man complx addv wh gaussan nos (AWGN. In SISO wrlss communcaon ssms Shannon channl capac s h maxmum muual nformaon bwn h ransmd sgnal and h rcvd on [], capac can b xprssd as W log ( (3 Whr P / h ransmd sgnal o nos rao and W s s h ransmsson bandwdh. Th channl s powr lmd n h sns ha p E{ x }. In cas of mulannna communcaon ssms (SIO, mulpl npu sngl oupu (ISO and IO, h rgodc capac s dfnd as ([9], [] E{ W log ( h H h } ( whr E{.} dnos h xpcaon opraor whch s valuad ovr h pdf of channl gan marx, h. Hr H dnos h complx conjuga ranspos. For SIO [], h s a vcor of sz n, hn n H h h h (5 Th rgodc capac bcoms n h E{ W log ( } (6 Usng small argumn lm approxmaon [], h zrohordr modfd Bssl funcon of h frs knd can b approxmad as I Accordng o abov approxmaon h Nakagam-q dsrbuon gvn n ( bcoms whr r Undr hs approxmaon h zroh ordr modfd Bssl funcon of Nakagam-q dsrbuon ( bcoms ( q ( q I q ( q Usng h abov approxmaon Nakagam-q dsrbuon gvn ( bcoms ( p ( ( q ( q q q ( q Equaon ( s smplfd as follows p ( (9 ( ( Whr ( q ( q, and ( q ( q Equaon ( s Nakagam-q dsrbuon undr larg lm argumn approxmaon.. POBABILITY DISTIBUTION FUNTION ALULATION Th followng PDF calculaon s shown for small lm argumn approxmaon. Accordng o [9], h rgodc capac of sngl-npu mulpl-oupu (SIO ssms rqurs h xac knowldg of h PDF of h subchannl gans. In (6, h PDF of nvlop of h dno b n Nakagam-q dsrbuon (. Now dfn n h (3 W hav ~, whr n n varabl wh n dgrs of frdom. s a cnral ch-squar Th dsrbuon of consdrd undr h followng wo cass: ( q ( q p (, (7 q Equaon (7 s Nakagam-q dsrbuon undr small lm argumn approxmaon. If w s fadng paramr q= n quaon (7 hn bcoms algh fadng dsrbuon. Now w wll drv h xprsson for Nakagam-q dsrbuon usng larg argumn approxmaon. Accordng o [3], h zroh ordr modfd Bssl funcons can b wrn as I r r (8 r. as A (Non- dncall ndpndn Nakagam-q In hs cas, h dsrbuon [] of s approxmad b: whr p (!, ( ( n, (, ( and n T q n T (5
DUET Journal Vol., Issu, Jun q and ar h h subchannls fadng paramr and sgnal o nos rao rspcvl.. as B (Idncall ndpndn Nakagam-q For hs cas all subchannls hav dncal gans and q = q, and qn. Th dsrbuon of s approxmad b: qn p ( ( qn!, ( (6 Whr ( ( qn and ( ( qn ( qn ( qn (7 5. ANALYTIAL EXPESSION FO SIO HANNEL EGODI APAITY Th followng capac calculaon s shown for small lm argumn approxmaon. In hs scon w wll drv analcal xprssons for dncall ndpndn as wll non-dncall ndpndn Nakagam-q fadng SIO channls. Th rgodc capac [9] of h ssm can b oband b usng (, p( log ( d p( / ln( = d (8 ln Subsung ( n o (8 and h dncall ndpndn Nakagam-q fadng rgodc capac bcoms: nond ln( d!ln (9 For dncall ndpndn Nakagam-q fadng cas, subsung (6 n o (8 h rgodc capac bcoms: qn d ln( d qn ( qn!ln ( Th analcal xprsson [] for h SIO channl capac from Equaon ( s gvn b followng horm. Thorm Whn compl channl sa nformaon (SI s avalabl a h rcvr, h SIO rgodc capac for algh fadng channl can b xprssd as a funcon of h numbr of rcvr annnas and h SN a ach rcv annna. Is analcal xprsson s ErgSIO, ln! k k E ln k l k k! k l! ( Whr k! s h facoral of a nonngav ngr k. E x s h xponnal ngral. Equaon ( s h xac closd-form capac xprsson for SIO ssms whch avods numrcal ngraons or on arlo smulaons. In pracc, h valus of could b calculad n advanc and sord n lookup abls. Prov of horm In PDF calculaon as w mnond arlr ha h probabl dsrbuon funcon (PDF of s p ;! Thus, log log E p d Whr (a / ln( d!ln (b / ln( d = / / d ln( d / / d d 3 / ln( d!! / d! / ln( d (c Equaon (c can b dvdd n o wo pars: / / / d d / k k k / k d k k k l / E and k l / / / ln( d d / E k! k l (d (!
apac[bs/sc/hz] apac [Bs/Sc/Hz] DUET Journal Vol., Issu, Jun Equaons (a-( can b combn o gv quaon (. To valua h rgodc capac gvn n non-dncall ndpndn Nakagam-q fadng cas (9 and dncall ndpndn Nakagam-q fadng cas ( w can us followng rsul from quaon (: ln( d ( (! (! k k l (! ( (/ E( k l k( k!( k l! whr E ( x s h xponn ngral dfnd as, ( ( E x d (3 x B usng (, h rgodc capac of Non- dncall ndpndn Nakagam-q (9 s approxmad b: nond!ln! / E / /! k l k ( /! / k l k k!( k l! ( Th rgodc capac for dncall ndpndn Nakagam-q fadng subchannls ( s approxmad b: d qn ( qn!ln qn ( qn! qn / E / qn / ( qn! qn qn k l qn qn k ( qn ( /! qn / k l k( qn k!( k l! (5 6. SIULATION AND ESULTS In hs scon, w wll nvsga h varaon of h rgodc channl capac wh rspc o varous paramrs for non-dncall ndpndn and dncall ndpndn dsrbud channls. For all of h gvn fgurs n hs papr rgodc channl capac has bn shown agans h ransmd sgnal o nos rao and numbr of rcvrs. From Fg., for all (cas, cas and cas 3 cass s obvousl found ha rgodc capac of non-dncall ndpndn Nakagam-q fadng channls s ncrasng along wh h ncrasng ransmd sgnal o nos rao. 9 8 7 6 5 3 as as as 3 5 5 5 3 SN [db] Fg.: Ergodc capac for non-dncall ndpndn Nakagam-q subchannls wh rspc SN [db] for varous cass. In Fg. cas rprsns non- dncall ndpndn Nakagam-q wh hr subchannls wh fadng paramrs q=.55,.55,.55 and = 3, 5, 7. In hs cas capac a db s 5.8 bs/sc/hz and ncrasng capac a 3 db s found 8.76 bs/sc/hz. In Fg. cas rprsns non-dncall ndpndn Nakagam-q wh hr subchannls wh fadng paramrs q=.5,.5,.5 and = 3, 5, 7 and capac a 3 db s found 5.35 bs/sc/hz. In Fg. cas 3 rprsns non-dncall ndpndn Nakagam-q wh hr subchannls wh fadng paramrs q=.5,.5,.5 and =, 3, 5. In cas 3 rgodc capac a db s.6 bs/sc/hz and capac ncrass along wh ncrass and a 3 db capac s. bs/sc/hz. Ths dffrn cass of smulaon show ha our drvd quaon s workng prfcl. For ach cas w found h sam rsul ha w xpc. Tha s capac of h ssm ncrass wh ncrasng ransmd sgnal o nos rao. 9.5 9 8.5 8 7.5 7 6.5 =3 =5 6 =7 =9 5.5 5 5 5 3 SN[dB] Fg. : Ergodc capac for h dncall ndpndn Nakagamq subchannls wh rspc o SN [db] for dffrn valus of.
apac[bs/sc/hz] DUET Journal Vol., Issu, Jun 9 8.5 8 7.5 7 6.5 6 ρ = 5 db ρ = db ρ = db ρ = 5 db ρ = db = db =5 db = db 5.5 =5 db = db 5 3 5 6 Numbr of cvrs Fg. 3: Ergodc capac for h dncall ndpndn Nakagamq subchannls wh rspc o numbr of rcvrs for dffrn valus of ransmd sgnal o nos raos. Th smulaon rsuls prsn n Fg. shows rgodc capac of h dncall ndpndn dsrbud. Nakagam-q subchannls wh rspc o avrag SN. In hs smulaon fadng paramr s s o.85 and numbr of rcvng annnas, n =. For dffrn valus of rgodc capac s shown. As h valu of ncrass h capac ncrass along wh dffrn valus of ransmd SN. So, avrag SN s a funcon of channl capac. Fg. 3 rprsns rgodc capac for h dncall ndpndn Nakagam-q subchannls wh rspc o numbr of rcvrs for dffrn valus of nsananous ransmd sgnal o nos raos. For hs smulaon, w usd fadng paramr.5 and = 9. I s obsrvd ha as h numbr of rcvrs ncrass for ach valu of nsananous ransmd SN h rgodc capac of h ssm ncrass. So, numbr of rcvr s a funcon of channl capac. 7. oncluson In hs papr, w hav drvd analcal xprssons for h rgodc capac of boh..d. and non-dncall ndpndn Nakagam-q fadng channls usng small argumn approxmaon of zroh-ordr modfd Bssl funcon of frs knd. W prsn numrcal and smulaon rsuls for boh..d. and non-dncall ndpndn Nakagam-q fadng sngl npu mulpl oupu channls. I s found ha h rgodc capac of h Nakagam-q fadng subchannls s proporonal o h ransmd sgnal o nos rao of h subchannls. EFEENES [] Z. Yf, Z. ng, X. Lmn, and W. Jng, Analcal xprsson for h IO channl capac, Tsnghua Scnc and Tchchnolog,, no. 3, pp. 7 77, 6. []. Shannon, ommuncaon hor of scrc ssms, Bll Ssm Tchncal Journal, 9, pp. 656 75, 99. [3] opoks, G.A.; onoganns, A.A.; ahopoulos, P.T, Informaon a ouag probabl of orhogonal spac-m block cods ovr Ho dsrbud fadng channls, Inrnaonal Workshop on Sall and Spac ommuncaons, pp. 35-39, 6. [] S. A. Fasolo,. S. Duqu. Fadng channl smulaor for Ho dsrbuon, Insuo Naconal d Tlcomuncacos, Brasl. [5] Fradnrach, Gusavo and Lévêqu, Olvr and off, John., On h IO channl capac for h dual and asmpoc cass ovr Ho channls, IEEE ommuncaons Lrs, Vol., pp.3-33, 7. [6] Youssf, N.; Wang,.-X.; Pazold,., A sud on h scond ordr sascs of Nakagam-Ho mobl fadng channls, IEEE Transacons on Vhcular Tchnolog, Vol. 5, pp. 59-65, 5. [7] X. Lu, Opmzaon of Sall Opcal Transmsson Subjc o Log-Squar-Ho Fadng, IEEE Inrnaonal onfrnc on ommuncaons (I,. [8]. K. Smon and. S. Aloun, Dgal ommuncaon Ovr Fadng hannls, nd don, Nw York. Wl,. [9] A.. agablh and.. aalgah, apac of SIO ssms ovr non-dncall ndpndn nakagam-m channls, n Proc. IEEE Sarnoff Smposum, pp. 5, 7. [] J. G. Proaks, Dgal ommuncaon Ssms, h don, cgraw-hll,. []. Z. I Sarkar, and T. anarajah, On h scrc muual nformaon of Nakagam-m fadng SIO channl, Inrnaonal onfrnc on ommuncaons (I, IEEE,. [] d. S. Islam and.. Islam Posv scrc muual nformaon ovr non-dncall ndpndnl dsrbud Nakagam-q fadng wrlss channl, Inrnaonal onfrnc on Engnrng, sarch, Innovaon and Educaon, p.77-8, 3. [3] K. S. Kolbg, Dfn ngral wh modfd Bssl Funcons, EN 6-5, Gnva, 96. []. Z. I. Sarkar, and T. anarajah, On h scur ouag prformanc for wrlss mulcasng hrough slow fadng channl. IEEE Inrnaonal onfrnc on ommuncaons (I,. 3