International Journal of Multidisciplinary Approach and Studies. Channel Capacity Analysis For L-Mrc Receiver Over Η-µ Fading Channel

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1 Chael Capaciy Aalysis For L-Mrc eceiver Over Η-µ Fadig Chael Samom Jayaada Sigh* Pallab Dua** *NEIST, Deparme of ECE, Iaagar, Aruachal Pradesh-799, Idia **Tezpur Uiversiy, Deparme of ECE, Tezpur, Assam, Idia ABSTACT: I his paper we used η-µ fadig chael as a fadig model ad maximal raio combier (MC) which is oe of he diversiy combiig echique cosidered a he receiver. Closedform expressios for he capaciy of maximal-raio combiig (MC) diversiy sysems over η-µ fadig chael are obaied ad aalyzed for a arbirary umber of ipu braches. Chael capaciy for adapive rasmissio echiques: Cosa power wih opimum rae adapaio (OA), Chael iversio wih fixed rae (CIF) ad Trucaed chael iversio wih fixed rae (TIF) are derived. The effec of diversiy order ad fadig parameers o he chael capaciy wih differe adapive rasmissio schemes has bee sudied. Keywords:ƞ -µ disribuio, MC receiver, Opimum rae adapaio (OA), Trucaed chael iversio wih fixed rae (TIF), Chael iversio wih fixed rae (CIF) INTODUCTION Capaciy aalysis of fadig chaels makes impora role i desigig ad implemeaio of wireless commuicaio sysems ad o improve specrum efficiecy ad service qualiy hereby providig useful iformaio []. For small scale fadig pheomea where here is o lie of sigh compoe, ƞ -μ chael model is used. Besides ohers commoly used fadig models such as ayleigh, Nakagami-m, Nakagami-q ec., ca be realized as he special case of ƞ -μ chael model [].Mohamed-Slim Alouii el. a [] sudied he Shao capaciy of adapive rasmissio echiques i cojucio wih diversiy combiig. This capaciy provides a upper boud o specral efficiecy usig hese echiques. Closed-form soluios for he ayleigh fadig chael capaciy uder hree Page : 8

2 adapive policies: opimal power ad rae adapaio, cosa power wih opimal rae adapaio, ad chael iversio wih fixed rae are obaied. Performace of a L brach maximal raio combiig (MC) receiver are aalyzed i equally correlaed η-μ fadig chaels. Mahemaical expressios for he PDF, momes, ouage probabiliy ad ABE for biary, cohere ad o-cohere modulaios are preseed i [4]. I paper [5], a umber of ew closed-formexpressios for he η-μ fadig chaels ivolvig he joi saisics of he evelope, phase, ad heir ime derivaives are obaied. A umber of ew exac secod order saisics for he η-μ fadig chaels are derived. The res of his paper is orgaized as follows. I Secio II, he iroducio of η-μ disribuio is give ad i Secio III he capaciy of MC combier sysem is discussed. I Secio IV, umerical aalysis ad resul have bee give. Fially, he paper is cocluded i Secio V THEȠ-µ DISTIBUTION.The ƞ-µ disribuio is a geeral fadig disribuio ha ca be used o beer represe he small-scale variaio of he fadig sigal i a o-lie-of-sigh codiio which may appear i wo differe formas. However, i mahemaical erms, oe forma ca be obaied from aoher by he relaio: Forma Forma Forma Where << ƞ i Forma. Forma is he parameer ƞ i Forma, ad - << is he parameer. ƞ -µ Disribuio: Forma I Forma, < ƞ < is he scaered-wave power raio bewee he i-phase ad quadraure compoes. I such case, ad H. I is oed ha wihi <ƞ, we have H. 4 O he oher had, wihi <ƞ -, we have H. Because v v I z I z,he disribuio yields ideical values wihi hese wo iervals, i.e., v h 4 i is symmerical aroud ƞ =. Therefore, as far as he evelope (or power) disribuio is cocered, i suffices o cosider ƞ oly wihi oe of he rages. We oe ha i Forma, H / h / +. Page : 9

3 . ƞ -µ Disribuio: Forma I Forma, - <ƞ < is he correlaio coefficie bewee he scaered-wave iphase ad quadraure compoes of each cluser of mulipah. I such a case, h ad H. We oe ha wihi ƞ <, we have H. O he oher had, wihi - <ƞ, we have H. Because I z v I z, he disribuio yields ideical values wihi hese wo v v iervals, i.e. i is symmerical aroud ƞ =. Therefore, as far as he evelope (or power) disribuio is cocered, i suffices o cosider ƞ oly wihi oe of he rages. We oe ha i Forma, H/h = ƞ.. CAPACITY OF MC COMBINE SYSTEM Iiially, cosider he physical model for he η-µ disribuio Forma. The evelope, ca be wrie i erms of he i-phase ad quadraure compoes of he fadig sigal as l i i i ( X Y ) where X i ad Y i are muually idepede Gaussia processes wih, E X EY i, E Xi, ad X EY i,ad is he umber of clusers of mulipah. Now Y so ha. Xi Yi i i i Usig his chael model he PDF of oupu SN for L-MC receiver is obaied which is give by f ( ) ( ) ( ) ( ( L ) ) ( ) L( ) L ( ) ( ) e ( )( ( L ) )! ( L ) L ( ) F L ; L ; ( ) Where fucio [6] ad x ( ad Γ( ) is he gamma fucio, a () F.;.;. is he coflue hypergeomeric xa is he Pochhammer s symbol [8,(6..)].Wriig he x Page : hypergeomeric fucio i erms of ifiie series, we have

4 f Where, ( ) F ( L ; L ; ) ( ) ( ) ( )! ( ) ( ) ( ) L. Cosa Power wih opimum rae adapaio The formula for OA is give by [7] L ( ( L ) ) ( ),,!!! ( L ) ( ) L ( ) L( ) ( ) ( )( ( L ) ) ( ) L e ( L ) ( L ) 4 By puig he values of f ( ) i he formula we obai C B log ( ) f ( ) d 5 OA C OA L ( ) L ( ) ( ) ( ) ( ( L ) ) ( ),,!!! L L( ) ( L ) L ( )( ( L ) ) ( ) log ( ) ( ) earragig he equaio we ca wrie i as C OA ( ) ( ) IL e L d log e L L( ) ( L ) L ( )( ( L ) ) ( ) ( ) ( ) ( ) ( ( L ) ) ( ),,!!! 6 7 Page : ( ) I l e d

5 Where ad is evaluaio give i he Appedix A. The value is give by. Puig he above value of i (7) ad rearragig he equaio, we have C OA ( k, ) I ( ) ( )! e K L ( ) ( ) ( ( L ) ) ( ) log el (L )!,,!!! ( L ) L k ( ) ( ) L ( )( ( L ) ) ( ). Trucaed chael iversio wih fixed rae The capaciy for his scheme is give by Cifr Blog P 9 ifr Where ifr f d ou ad ou P f d For his, a soluio o he iegral i ifr ad P i [7] is give. Usig (), his ca be obaied by solvig he resulig iegral usig [8, (.8.)]. The expressio for foud ou i he followig maer. By puig he value of ou f i (9), we have ifr ca be ifr ( ) ( ) ( ( L ) ) ( ) ( ) ( ) L The above expressio ca be expressed as L L L( ),,!!! ( L ) ( )( ( L ) ) ( ) (L ) ( ) e d Page :

6 ( ) ( ) ifr ( ( L ) ) ( ) L ( ) L!!! ( ) ( )( ( L ) ),, L ( ) L, ( ) L L ( ) ( ) ( ) L ( ) v x v We have x e dx v, u u, e u The fial expressio afer simplificaio for,, ifr ca be give as ( ) L, L ( ) h Ouage probabiliy ca be defied as pou f mrc dmrc, where h is a hreshold value of he oupu SN [7]. Usig (9), he resulig iegral ca be solved by expressig he hypergeomeric fucio i ifiie series ad usig [8, (.8.)]. The fial expressio for he ouage probabiliy is give as p ifr L L L ( L ) mrc ou L ( ( L ) ),,!!! L ( ) ( ) ( ( L ) ) ( ) L ( ) L!!! ( L ) ( ( L ) ) ( ) g L, () ( ) N x ad a is he icomplee gamma fucio [5, (6.5.)] g a, x e d ad is he ormalized average brach SN. The fial expressio afer simplificaio h ca be give as Page :

7 p L ou L ( ( L ) ),,!!! L L ( L ) ( ) L, ( ) L The fial expressio for capaciy for TIF scheme ca be obaied by puig he values of ad ifr P. ou. Chael iversio wih fixed rae The capaciy for his scheme is give by 4 C Blog 5 For his scheme requireme o fid a soluio o he iegral. I ca be solved by puig () ad he solvig he resulig iegral usig [8, (7.6.4)]. The procedure is show below. The formula for is give by f d o.puig () i his formula we have ( ) ( ) L ( ) L( ) L L ( )( ( ) )! ( ) ( ) F L ; L ; d ( ) L ( ( ) ),! Usig he formula [8, (7.6.4)] ( ) L ( ) e 6 b ; ; ; ; ; e F a c k d b s F a b c ks s b s k The give iegral ca be solved as Where F a, b; c; z is he hypergeomeric fucio. The expressio afer algebraic maipulaio Page : 4

8 ad simplificaio ca be give as ( ) ( ) ( ( L ) ) ( ),! L (L ) ( ( L ) )! ( L ) ( ) F L,L ; L ; 8 Thus he fial expressio for he capaciy of his scheme ca be obaied by puig (8) io (5).,! ( )( ( L) )! ( L ) ( ) L ( ) ( ) ( ( L ) ) ( ) L( ) (L ) ( ) L ( ) ( ) F L,L ; L ; ( ) ( ) ICAL ESULT AND DISCUSSION 7 NUME 4.The expressios for capaciy wih differe power ad rae adapaio echiques are obaied. These expressios are umerically evaluaed for differe values of fadig parameer ad diversiy order ad ploed for illusraio. Capaciy (per ui badwidh) of OA scheme has bee ploed i Fig.. I ca be observed from he figure ha for a give fadig parameer he capaciy icreases wih icrease i L. As he parameers η ad µ icrease he capaciy icreases i a liear fashio for low correlaio co-efficie i.e. ρ=.. For higher value of ρ he case for icrease i capaciy wih he icrease i he umber of brach is o saisfied which i ur is o saisfied which is show i fig.. The capaciy vs average SN for TIF ad CIF schemes has bee ploed i Figs. ad, respecively. I boh schemes i ca be observed ha capaciy icreases wih he icrease i L. The same case ha capaciy icreases wih he icrease i parameers η ad µ ca be observed agai for cerai ieresed value of ρ. A good chael capaciy i obaied whe he parameer ρ=. i Page : 5

9 boh he cases. I he plo of TIF scheme is assumed o be db. Therefore, plos are give for db owards. Capaciy plos for OPA scheme have o bee icluded here, bu i is possible o plo he capaciy from he give aalyical expressio. The umerical resuls obaied are verified agais he special case published resul ad foud o be machig. The covergece of he ifiie series ivolved i he obaied expressios has bee verified. Fig. Capaciy for OA scheme wih ρ=.fig. Capaciy for TIF scheme wih ρ=. Fig. Capaciy for CIF scheme wih ρ=.fig.4 Capaciy comparisos for OA, CIF ad TIF CONCLUSION 5. I his paper, he capaciy of L-MC diversiy sysem over η-µ Fadig Chael is aalyzed, for differe kow power ad rae adapaio rasmissio echiques. The various Page : 6

10 expressios for respecive adapive rasmissio echiques are obaied. Numerical evaluaios are carried ou for respecive schemes for he differe parameers L, η, µ ad ρ. The resuls are ploed for differe parameer of ieres ad compared wih he available special case resuls. I is observed ha Diversiy echique icreases he chael capaciy for all rasmissio schemes. Ou of he adapive rasmissio schemes, he maximum diversiy gai is observed i Opimum ae Adapaio mehod. APPENDIX A EVALUATION OF INTEGAL I 6.We evaluae he iegral I defied usig parial iegraio, amely u dv = lim uv lim uv v du A. Firs, le d u l, du A.. dv e d A Performig - successive iegraio by pars yields [.eq. (..), p. ] k k! v e A.4 k! Subsiuig (A.) ad (A.4) i (A.), we see ha he firs wo erms go o zero. Hece k k! e I d A.5! k k The iegral i (A.5) ca be wrie i a closed form givig k, I! e A.6 k k Where.,. is he complemeary icomplee gamma fucio. EFEENCES Page : 7

11 [] Adrea Goldsmih, Wireless Commuicaios, Saford Uiversiy, Cambridge Uiversiy Press. [] Michel DaoudYacoub, The η-μ Disribuio: A Geeral Fadig Disribuio DECOM/FEEC/UNICAMP C.P. 6 [] Mohamed-Slim Alouii, Member IEEE, ad Adrea J. Goldsmih, Member, IEEE Capaciy of ayleigh Fadig Chaels Uder Differe Adapive Trasmissio ad Diversiy-Combiig Techiques, IEEE Trasacios o Vehicular Techology, Vol. 48, o. 4, July 999. [4]. Subadar, Sude Member, IEEE, ad P.. Sahu, Sude Member, IEEE, Performace of a L-MC eceiver over Equally Correlaed η μ Fadig Chaels, IEEE rasacios o wireless commuicaios, Vol., o. 5, May. [5].A. Scholz, The Spread Specrum Cocep, i Muliple Access, N. Abramso, Ed. Piscaaway, NJ: IEEE Press, 99, ch., pp. -. D. B. da Cosa, J. C. S. S. Filho, M. D. Yacoub, ad G. Fraideraich, Secod-order saisics of η μ fadig chaels: heory ad applicaios, IEEE Tras. Wireless Commu., vol. 7, o., pp Mar. 8. [6] M. Abramowiz ad I. A. Segu, Hadbook of Mahemaical Fucios, Naioal Bureau of Sadards, 97. [7] A.J. Goldsmih ad P. P. Varaiya, Capaciy of fadig chaels wih chael side Iformaio, IEEE Tras. Iform. Theory,. vol. 4, pp , Nov. 997 [8] I. S Gradshey ad I. M. yzhik, Table of Iegrals, Series, ad Producs, 6h Ed., Sa Diego, CA: Academic,. Page : 8