On the Distribution of the Weighted Sum of L Independent Rician and Nakagami Envelopes in the Presence of AWGN

Transcription

1 On the Dstbuton of the Weghted Sum of L Indeendent Rcan and Naagam Enveloes n the Pesence of AWN eoge K Kaagannds and Stavos A Kotsooulos Abstact: An altenatve, unfed, sem-analytcal aoach fo the evaluaton of the cumulatve dstbuton functon cdf) of the weghted sum of L ndeendent Rcan o Raylegh as a secal case) and m-naagam enveloes wth o wthout the esence of Addtve Whte aussan Nose AWN) s esented The cdf s evaluated dectly n a nested mode va the Hemte numecal ntegaton technque The oosed fomulaton avods the calculaton of comlex functons and can be effcently aled to actcal weless alcatons when L 3, usng abtay statstcal chaactestcs fo the modelng aametes Moeove, t can be also used to contol the accuacy of othe technques when L>3 Comments, comason wth othe exstng technques and useful cuves fo seveal actcal weless alcatons such as the calculaton of the eo bounds fo codng on fadng channels n moble satellte alcatons and the Equal an Combnng EC), ae also esented Fnally, the elaton between the dstbuton of the sum of m-naagam and Rce enveloes s nvestgated and dscussed Index Tems: Rcan fadng, Naagam fadng, dvesty, equal gan combnng, moble satellte communcatons I INTRODUCTION In seveal actcal weless alcatons whch nvolve Rcan o Raylegh as a secal case) and Naagam fadng, thee s a need fo the calculaton of the cdf o the comlementay cdf ccdf) of the sum o geneally the weghted sum) of L statstcally ndeendent andom vaables RVs) wth o wthout the esence of AWN Such weghted sums occu n the calculaton of the eo bounds fo codng on Rcan fadng channels n moble satellte alcatons n whch the cdf of the weghted sum of L statstcally ndeendent Rcan RVs needs to be calculated [] Othe motant alcatons nvolve the evaluaton of the eo efomance n equal gan and maxmal ato combnng systems, sgnal detecton, lnea equalzes, outage obablty, ntesymbol ntefeence, and hase jtte One soluton to ths oblem s elated to the extacton of a smle exesson fo the chaactestc functon chf) of the sum Manusct eceved Setembe 4, ; aoved fo ublcaton by Yong Hoon Lee, Dvson I Edto, May 7, K Kaagannds s wth the Insttute of Sace Alcatons & Remote Sensng, Natonal Obsevatoy of Athens, eece, e-mal: gaag@sacenoa g S A Kotsooulos s wth the Weless Communcatons Lab, Electcal & Comute Eng Det, Unvesty of Patas, eece, e-mal: otso@eeuatas g 9-37$ c KICS o the weghted sum) of L ndeendent Rcan o m-naagam fadng amltudes, whch has not been solved adequately yet It must be noted hee that the devaton of the chf n Rcan fadng channels s moe comlcated comaed to the Naagam fadng case, because the Rcan obablty densty functon df) contans an exlct tem of the modfed Bessel functon of the fst nd [] In the evous yeas, many eseaches ted to meet ths need usng seveal technques Scannng the lteatue we can fnd seveal attemts fo the extacton of the cdf o the df of the sum of RVs elated to weless alcatons), but they ae lmted to sne waves and Raylegh RVs [3] [6] and they do not nvestgate the case of the weghted sum The most well nown aoach fo the Naagam fadng was made by Beauleu and Abu-Dayya [7], who used the nfnte sees eesentaton esented n [8] to obtan the ccdf Ths s an motant esult snce the oosed sees aoxmaton s geneal and smle comaed to evous ublshed technques Othe authos as Aloun and Smon [9] oosed anothe aoxmate aoach fo Naagam fadng channels usng Hemte numecal ntegaton Recently, the same authos obtaned closed-fom exessons fo the aveage sgnal-to-nose ato SNR) ove dvesty aths wth exonentally decayng owe delay ofle [] The authos of the esent ae oosed [] an altenatve aoach fo the evaluaton of the eo obablty ERRP) n Naagam fadng EC systems, whch s effcent fo low ode dvesty Othe eseaches who ted to coe wth the oblems asng n weless alcatons n whch the sum of Lm-Naagam RVs s nvolved ae lsted n [], [3] A comehensve summay fo the efomance analyss of dgtal communcatons technques ove genealzed fadng channels can be found n [4] As fa as the Rcan fadng s concened, Abu-Dayya and Beauleu [5] have oosed a method smla to [7] fo the evaluaton of the ERRP fo EC dvesty n Rcan slow fadng envonment fo coheent BPSK and non-coheent BFSK Late, the same authos examned the efomance of MPSK n the esence of co-channel ntefeence fo EC n Naagam and Rcan fadng envonments [6] Recently, Annamala et al n [] esented a dect technque exessed n tems of sngle o double fnte ntegals fo the evaluaton of the ERRP of EC systems n Raylegh, Rcan, and Naagam fadng channels Othe eseaches such as Zhang n [5], [6], esented a smle aoach fo the evaluaton of the ERRP fo coheent and non-coheent modulaton schemes n slow Raylegh fadng

2 channels usng the l-palaez lemma Fnally, ecently the authos of ths ae esented a method fo the calculaton of the ERRP n Raylegh fadng EC weless alcatons, whch s vey effcent comaed to othe technques, esecally when the numbe of dvesty banches s lowe than fou [7] It must be noted hee that to ou nowledge the soluton to the oblem of extactng the cdf of a weghted sum of fadng amltudes s lmted only to the Raylegh case n [8] In ths ae, we attemt to extact the cdf of the weghted sum of Lm-Naagam and Rcan RVs n the esence o not of addtve uncoelated whte nose avodng the calculaton of comlex functons and usng abtay values fo the modelng aamete The oosed sem-analytcal aoach assumes ndeendent fadng enveloes, whch ae not necessaly dentcally dstbuted The cdf of the weghted sum of two RVs s evaluated dectly usng the defnton and the oetes of the chf of a RV and a fomula whch can be evaluated va the Hemte numecal ntegaton method s deved Then, ths fomula s used n a nested mode fo the devaton of the cdf of the sum of L RVs Although the oosed aoach needs the evaluaton of well nown tabulated functons, ts nested fom maes t comutatonally cumbesome fo In such cases, t s useful to be used as a efeence ont n ode to contol the accuacy of othe technques On the othe hand, fo the oosed n ths ae method s vey effcent and smle comaed to othe technques Hence, t can be easly used fo alcatons nvolvng a low numbe of fadng RVs But, n any case, to ou nowledge, esults fo the dstbuton of the weghted sum of Rce and m-naagam fadng RVs, whch ae not dentcally dstbuted dffeent owes), have not been ublshed evously In Secton II, the oblem of the Rcan fadng case s fomulated wth the necessay mathematcal analyss and the fnal exessons fo the cdf wth o wthout the esence of AWN ae esented In Secton III, the oblem fo the m-naagam fadng case s solved In Secton IV, comments ae made and comute esults fo actcal weless alcatons llustate the oosed fomulaton In the same secton, the elaton between the dstbuton of the sum of m-naagam and Rce enveloes s nvestgated and dscussed Fnally, Secton V esents the ae s concludng emas II THE DISTRIBUTION OF THE WEIHTED SUM OF L RICIAN RVs Let xx x L be the amltudes of L statstcal ndeendent enveloes, whch follow the well-nown Rce dstbuton f RICE x x )= ex ; x + u I x u x whee x s the sgnal amltude, I s the zeo-ode modfed Bessel functon of the fst nd, s the aveage fadng- scatte comonent and u the lne-of-sght LOS) owe comonent The Local Mean Powe LMP) s defned as = + u and the Rce facto K of the -th enveloe s defned as the ato of the sgnal owe n domnant comonent ove the scatteed owe, e, K = u When K goes to zeo, the ) channel statstc becomes Raylegh, wheeas, f K goes to nfnty, the channel becomes a nonfadng channel Values of Rce facto n outdoo and ndoo systems usually ange fom to 5 [8], [9] Let L = x + cx + + c L x L be the vaable whch eesents the weghted sum of the L Rcan RVs wth c c L beng constants Fst, the cdf of the sum of the vaables x and cx denoted as wll be fomulated and then, ths sum wll be used n a nested mode fo the devaton of the cdf of L Let s) c x s), and c x s) be the chfs of the vaable x, and cx esectvely Then, due to the ndeendence between x and x s) can be wtten as [] s) =C s)c s) = s) cs) ) o fom the defnton of the chf Z s) = s) ex jcst) f t) dt 3) wth ft) beng the Rcan df of the RV x Usng ) and 3) and afte the tansfomaton t = s) can be wtten as s) = s)e Z ;K ex The cdf of x s defned as o e ; I jsc d: K 4) F v) = Pob [ v] 5) Z v F v) = f ) d Z v Z = s)ex;js) dsd whee s anothe auxlay vaable Now, usng 4) and 6) and tang nto account the fact that by defnton Z ; c s) ex ;js ds! 7) c =f ; the cdf of can be wtten followng a staghtfowad ocedue as Z F v) = e ;K F v ; c 8) I K ex; ; d wth Fv) beng the cdf of the x A fomulaton fo the cdf of the cx, when x follows Rcan df and c s constant, s easly 6)

3 found to be elated to the Macum Q-functon o smly Q- functon) as v F v) =; Q K : 9) An effcent fomulaton fo the Macum Q-functon has been gven n [4] Ths fomulaton s used n ths ae fo the evaluaton of Fv) n 9) The second at of 8) can be calculated numecally wth desed accuacy usng the Hemte numecal ntegaton method [, 875] Hence, the fnal sem analytcal closed fom fo the calculaton of F can be wtten as n F v) =e ;K a F v ; z c z # I z K ) whee a z, and n ae the weghtng factos, the abscssas and the ode of the Hemte numecal ntegaton method, esectvely [] It s mentoned hee that only the ostve values of abscssas ae used, because the ntegals ae defned ove the ostve half axe Moeove, only the values whch satsfy the condton, v ; z c c > ae taen nto account n the summaton of ) Followng the same mathematcal analyss n a nested mode, the fomulae fo the cdf of the weghted sum of L ndeendent Rcan RVs ae shown below n F v) = e ;K a z I F v ; z c n F3 v) = e ;K3 a z I F v ; z z K z K3 3 c 3 n FL v) = e ;KL a z I FL; v ; z z KL L c L : ) The aoach oosed n ) can be used fo Rcan enveloes wth abtay values fo K and A The Weghted Sum of L Rcan Enveloes n the Pesence of AWN If the L ndeendent Rcan enveloes ae tansmtted ove an AWN channel, the total sum of the desed sgnals lus the nose can be wtten as L L PNL = c x + w ) whee x s the outut sgnal amltude and w eesents the comlex aussan nose, whch affects at the -th enveloe wth zeo mean and vaance N = The chf of the aussan vaate w s well nown and can be exessed as w s) = ex ; N 4 s 3) whch s the chf of a aussan df wth zeo mean and vaance N = Due to the ndeendence between desed sgnals and nose, the chf PN L S) of the enveloe PNL can be wtten as o fom 3) PN L s) = LY w s) = LY x s) 4) = PN L s) = NORM L=) s) s) 5) = wth L beng the total owe of the aussan nose Followng the same mathematcal analyss as n the case wthout nose, the cdfs of the weghted sum of L ndeendent Rcan enveloes n the esence of AWN ae shown below FN v) = e ;K n a z I LY z K h F NORM L=) v ; z n FN v) = e ;K a z I h F v ; z c n FNL v) = e ;KL a z I h FL; z K z KL v ; z L c L 6) wth F NORM L=) x) beng the well nown standad nomal cdf wth zeo mean and vaance L = III THE DISTRIBUTION OF THE WEIHTED SUM OF Lm-NAKAAMI RVs Naagam fadng m-dstbuton []) descbes multath scatteng wth elatvely lage delay-tme seads wth dffeent clustes of eflected waves Wthn any one cluste the hases of ndvdual eflected waves ae andom, but the delay tmes ae aoxmately equal fo all waves As a esult, the enveloe of each cumulated cluste s Raylegh dstbuted The m-naagam model s also often used to descbe the ntefeence cumulated fom the multle ndeendent Raylegh fadng souces, atculaly f these ae dentcally dstbuted same LMPs) []

4 Let xx x L be L statstcal ndeendent RVs, whch follow the well-nown Naagam m-dstbuton wth df gven by [] f x )= m ) m xm ; ;m ) ex ; m x ) x = L 7) whee ;x) s the amma functon, contols the sead of the dstbuton fo sgnal alcatons eesents the LMP) and m eesents the nvese nomalzed vaance Followng the same mathematcal analyss as n the Rce case descbed above n Secton II, the fomulae fo the cdf of the weghted sum of L ndeendent m-naagam RVs ae shown below F v) = F3 v) = FL v) = ;m) n a z m; F v ; z cc ; ;m3) n a z m3; F v ; z c3c ; ;m L ) n a z ml; FL; v ; z c L c ; m 3 m3 L m L wth Fv) beng the cdf of the cx, easly found to be F v) =P 8) m m c v 9) and P beng the ncomlete amma functon Ths functon can be evaluated usng a outne fom a mathematcal softwae acage eg, MATHEMATICA) The aoach oosed n 8) can be used fo andom m-naagam vaates wth abtay values fo and m aametes A The Weghted Sum of Lm-Naagam Enveloes n the Pesence of AWN In the esence of AWN, followng smla mathematcal analyss as n Secton II-A, the fomulae fo the cdf of the weghted sum of L ndeendent m-naagam RVs ae shown below FN v) = FN v) = FNL v) = ;m) n a z m; F NORM L=) ;m) FN ;m L ) n n a z m; v ; z c a z ml; v ; z m m FNL; v ; z c L L m L : ) IV PRACTICAL APPLICATIONS, COMMENTS, AND NUMERICAL RESULTS Some comments on the Beauleu and Abu-Dayya s technque esented n [7], [8], and [5] ae as follows: a) The need fo the evaluaton of secal comlex functons as the confluent hyegeometc esents seous tme oveflow oblems unde some ccumstances [3, Aendx B] Moeove, usng ths sees aoxmaton two eos occu [4] The fst ases due to the assumton of bounded andom vaables and the second due to temnaton of the nfnte sees These eos ae also dscussed n [4] b) The Naagam m-aamete s constaned to tae ntege values Ths s not tue fo eal moble ado envonments c) The accuacy of comutaton fo a secfc value deends stongly on the aoate selecton of the aamete T and the numbe of non-zeo tems n nfnte sees n [7], whch ae dffeent fo dffeent values of L and m In ode to show the geneal alcablty of the oosed technque, we assume that the enveloes have abtay values fo the statstcal aametes m and K We also assume that the LMPs follow unfom = = = L) o exonental owe decay ofle, gven by =e d;) = Lwth d beng the owe decay facto [4] A The Rcan Fadng Case In Fg, the cdf of the weghted sum of L Rcan enveloes s lotted vesus x=l :5 fo nomalzaton uoses as n [4]) fo seveal numbes of L, 3, and 4) and abtay values of Rce facto K ncludng the Raylegh case of K =) and fo the weghtng coeffcents In the same fgue, t s assumed that =fo all enveloes In Fg, the cdf s lotted vesus x=l :5 fo L = 3 and 4 and abtay values fo the weghtng coeffcents In ths fgue, t s assumed that a) the K facto s the same fo all enveloes and b) the LMP of each enveloe follows unfom d = ) o

5 Cumulatve Dstbuton Fx) & & & & assumed that x emans constant wthn symbol duaton, but changes fom symbol to symbol followng the Rcan df The aveage SNR at the -th banch s defned as = L =L ) wth L beng the total n all banches) owe of the aussan nose and the total sgnal owe-sum of the LOS and scatteed at the -th banch Hence, assumng fo smlcty that the Rce facto s the same fo all banches K = K = = K L = K) can be wtten as xl^5) Fg The cdf of the weghted sum of L Rcan enveloes fo abtay values fo the Rce K facto and the weghtng coeffcents =fo all enveloes K +) = : 3) L =L The Eo Pobablty ERRP) fo coheent BPSK detecton s defned as P e L) = Pob [ L) < ] 4) Cumulatve Dstbuton Fx) & & & & whch s equal to FNL), wth FNLv) defned n 6) Hence P e L) = L e ;LK a =j n Y =j n a j j= n= s a n L K z +) t I z K A 5) xl^5) Fg The cdf of the weghted sum of L Rcan enveloes fo abtay values fo the weghtng coeffcents and the owe decay facto d K =:8 fo all enveloes exonental decay ofle d 6= ) To the best of the authos nowledge, such esults have not been evously esented The cuves of Fgs and can be effcently used fo the calculaton of the eo bounds fo codng on Rcan fadng channels n moble satellte alcatons [] As efeed above, a actcal alcaton of 6) s n the eo analyss of EC systems In an EC system wth coheent detecton, the sgnals eceved n each banch ae co-hased, summed and coheently demodulated The decson vaable whch s assumed to be constant wthn symbol duaton) fo a coheent BPSK can be fomulated as L L L) = x + w ) = = whee x s the outut sgnal amltude at the -th banch and w eesents the comlex aussan nose at the -th banch wth zeo mean and vaance N = The sgn n ) s ostve f the tansmtted symbol equals one and s negatve f zeo It s whch s ndeendent to L and Qx) s the well-nown aussan Q-functon [4, 7] At ths ont, t must be mentoned hee agan that only the ostve values of abscssas ae used, because the ntegals ae defned ove the ostve half axes Also, t must be noted that fo the case of efect coheent detecton and no co-channel ntefeence, the ERRP efomance of BPSK s dentcal to that of QPSK In the case of coheent BFSK, the nose vaance s doubled comaed to the case of coheent BPSK Hence, the oosed fomulaton s also vald wth elaced by = n 5) In Fg 3, the ERRP fo coheent BPSK s dected as a functon of the SNR at the fst banch = fo seveal odes of dvesty L = 3), assumng that the sgnals avng at each banch have the same Rce facto K = 5 db We also assume that the LMPs and consequently the SNRs at each banch follow exonentally owe decay ofle In ths case = e ;d;) = Lwth d beng the owe decay facto Some comments and comason between the method oosed n [5] and the technque oosed n ths ae ae as follows: a) Eqs ) and 6) ae n a nested fom wth L ; summaton loos fo L enveloes wthout the esence of AWN) o L loos fo L enveloes n the esence of AWN) In ths case, whch s comaable to [5], each summaton loo conssts of non-zeo tems Hence, the non-zeo tems that need to be summed hee ae M = L The nfnte sees n [5, Eq 3a)] s also n a nested fom [5, Eqs 3a), 3b), 3c),

6 Eo Pobablty % % % Cumulatve Dstbuton Fx) & & & & 65ρ DW RQH EUDQFK % Fg 3 ERRP fo coheent BPSK, vesus SNR n one banch fo seveal dvesty odes L, decay facto d and Rce facto K xl^5) Fg 4 The cdf of the weghted sum of L Naagam enveloes fo abtay values fo the weghtng coeffcents and the owe decay facto m =:5 3 3:) =4) 4), 5)] wth thee summaton loos fo evey L, two of them beng nfnte sees and the thd conssts of L non-zeo tems If n s the selected numbe of the non-zeo tems that needs to be summed n nfnte sees [5, Eq 3a)] fo a secfc accuacy) and n s the coesondng numbe fo the nfnte sees of [5, Eqs 4), 5)], the total numbe of the tems needs to be summed s M = nln The numbe n deceases as the L nceases as efeed n [8] Fo L =the values fo M and M ae 56 and 67 and fo L =3the coesondng values ae 97 and 84 Fo L =4these values ae 565 and 78, esectvely The desed accuacy s chosen to be u to sxth dgt fo both methods It s obvous that the technque oosed n ths ae s an effcent altenatve to the nfnte sees concet of [5] and [8] fo L 3, as fa as the seed of comutaton s concened On the othe hand, fo L>3and fo hghe values of L t becomes comutatonally cumbesome and Dayya and Beauleu s technque seems to be a moe attactve tool fo the calculaton of the ERRP wth accuacy and seed, whch nceases as the L nceases In ths case L >3) and due to ts non-comlcated fom, the oosed n ths ae technque can be used fo accuacy comang uoses wth othe methods b) As fa as the comlexty of the calculatons s concened, the functons that needed to be evaluated n evey non-zeo tem usng Eqs ) and 6) ae the well-nown aussan Q-functon n the esence of AWN), the Macum Q-functon wthout AWN) and the zeo-ode modfed Bessel of the fst nd Q Howeve, the tem t I t K seems eo one, =j n snce Ix) gets lage qucly wth the ncease of ts agument, and the multlcaton of seveal lage numbes could lead to naccuate esults In ode to solve ths oblem, the authos oose the calculaton of e ;x Ix) usng an exanson n sees of Chebyshev olynomals [, 937] nstead of the dect calculaton of Ix) In ths case, the tem e ;LK must be ncluded n the summatons On the othe hand, the technque esented n [5] needs the evaluaton of two nds of the confluent hyegeometc functon n evey non-zeo tem [5, Eqs 4), 5)] The calculaton of such functons as t s also efeed n [3, Aendx B] esents seous tme oveflow oblems unde some ccumstances and the develoment of aoate methods s necessay n ode to avod them c) The eo that s ntoduced usng Eqs ) and 6) can be calculated usng the fomula fo the emande quantty of the Hemte numecal ntegaton method [, 5446] The total emande fo L =s the -ode summaton of the atal emandes and fo L =3 s the -ode coesondng summaton The eo that occus due to the temnaton of the nfnte sees of [5, 6] s dscussed n [4] B The m-naagam Fadng Case Eqs 8) and ) ae used fo the evaluaton of the dstbuton of the weghted sum of Lm-Naagam RVs wth o wthout the esence of AWN The accuacy of comutatons s unde contol n = s needed fo an accuacy of sx dgts) and the dsadvantage of lengthy comutaton tme ases when the numbe of RVs s geate than thee In Fg 4, the cdf s lotted vesus x=l :5 fo L = and 3, abtay values fo the Naagam m aamete, seveal values fo the weghted coeffcents and fo the owe decay facto d To the best of the authos nowledge, such esults have not been evously esented The technque oosed n ths ae s vey effcent fo small values of v snce, fo examle, the calculaton of F) needs the summaton of 9 non-zeo tems and the coesondng value fo F7) s The numbe of non-zeo tems that need to be summed fo the evaluaton of the nfnte sees of [7] unde the same accuacy demands) s about 45, wthout tang nto account the calculaton of the nvolved two tyes of confluent hyegeometcs functon The numbe of non-zeo tems that need to be summed fo F3) ae 8, and 4 fo F37) The coesondng numbe of non-zeo tems fo the method esented n [7] s about 4 Followng the same analyss as n Secton II-A fo the Rcan case, the ERRP fo BPSK EC systems n Naagam fadng

7 P P 5LFH DSSUR[LPDWLRQ E\ D DNDJDPL SI 5LFH SI 5LFH DSSUR[LPDWLRQ E\ D DNDJDPL FI 5LFH FI Eo Pobablty P P Pobablty x 65ρ DWRQHEUDQFK % Fg 6 Rce df and cdf fo K = 4:45 = 3and the coesondng Naagam aoxmaton usng the fomulae esented n [] Fg 5 ERRP fo coheent BPSK vesus SNR at a efeence banch fo seveal dvesty odes L, decay factos and Naagam m aamete channels s gven by P e L) = [; m)] L w =j n w j j= n= z Lm A w n Y =j n z m; 6) Cumulatve Dstbuton Fx) 5LFH D ND JD PL D SUR[ 5LFH D ND JD PL D SUR[ whee Qx) s the aussan Q-functon In 6), fo smlcty uoses, t s assumed that all of the enveloes have the same value fo the Naagam m aamete In Fg 5, the ERRP fo coheent BPSK s dected as a functon of the SNR at the fst banch = fo seveal odes of dvesty L = 3), assumng that the sgnals avng at each banch have the same m aamete m = 3:5) We also assume that the LMPs and consequently the SNRs at each banch follow exonental owe decay ofle wth d = :5, and C Aoxmaton of the Rcan cdf by the m-naagam cdf Sometmes the m-naagam model s used to aoxmate a Rcan dstbuton [5] because of ts fom smlcty comaed to the Rcan case Naagam n [, 7 8] has gven some fomulae fo the elaton between m-dstbuton and n-dstbuton Rce) In Fg 6, the Rce df and cdf wth K =4:45 and =3ae dected togethe wth an aoxmaton of ths dstbuton by a m-naagam df [, 7 8] Although, ths aoxmaton seems to be accuate fo the man body of the df and cdf), t becomes hghly naccuate fo the tals As BER manly occus dung dee fades, the tal of the df manly detemnes these efomance measues the coesondng cdf In Fg 7, the cdf of the weghted sum of L =and 3 Rcan RVs wth abtay values fo the K facto s aoxmated by the coesondng m-naagam one, usng the fomulae of [, 7 8] xl^5) Fg 7 Rce cdfs of the weghted sum of L RVs and the coesondng Naagam aoxmated cdfs fo abtay values of K: C :8 :4) =3d =:8 As t s aaent hee, the Naagam aoxmaton oveestmates the efomance of the equvalent Rcan model These esults ae n accodance to the comments, whch ae ncluded n [7, 65 66] Moeove, n [7] ths aoxmaton s dscussed n deth fo EC systems But, t has to be mentoned that ths oveestmaton s geate fo weghted sums and not unfom dstbuted enveloes Theefoe, n ou onon t s not coect to aoxmate the cdf of the sum of L Rcan enveloes sgnal by the coesondng m-naagam one f the alcaton s a BER analyss V CONCLUSIONS A novel, smle and flexble aoach fo the evaluaton of the cdf of the weghted sum of L Rcan and m-naagam statstcally ndeendent enveloes n the esence o not) of AWN has been analyzed and esented The obtaned fomulaton ovdes accuacy and seed fo L 3 and can be easly used n a wde ange of weless alcatons, whch nvolve Naagam and Rce fadng channels Equal an Combnng and calcula-

8 ton of the eo bounds fo codng on fadng channels n moble satellte alcatons) Moeove, when L > 3 t can also be used to contol the accuacy of othe technques Followng the same analyss, the oosed method can be adated to extact the cdf of the sum of the owes of Lm-Naagam o Rce dstbuted RVs Such a esult can be used n outage obablty analyss n cellula netwos REFERENCES [] D Dvsala and M Smon, Tells coded modulaton fo bs tansmsson ove a fadng moble satellte channel, IEEE J Select Aeas Commun, vol SAC-5, no, 6 75, Feb 987 [] A Annamala, C Tellambua, and V Bhagava, Unfed analyss of equal-gan dvesty on Rcan and Naagam fadng channels, n Poc IEEE WCNC 99, Set 999, 4 [3] S Rce, Pobablty dstbutons fo nose lus seveal sne waves The oblem of comutaton, IEEE Tans Commun, , June 974 [4] C Helstom, Comutng the dstbuton of sums of andom sne waves and of Raylegh-dstbuted andom vaables by saddle-ont ntegaton, IEEE Tans Commun, vol 45, no, , Nov 997 [5] Q Zhang, Pobablty of eo fo equal gan combnes ove Raylegh channels, IEEE Tans Commun, vol 45, no 3, 7 73, Ma 997 [6] Q Zhang, A smle aoach to obablty of eo fo equal gan combnes ove Raylegh channels, IEEE Tans Veh Technol, vol 48, no 4, 5 54, July 999 [7] N Beauleu and A Abu-Dayya, Analyss of equal gan dvesty on Naagam fadng channels, IEEE Tans Commun, vol 39, no, 5 34, Feb 99 [8] N Beauleu, An nfnte sees fo the comutaton of the comlementay obablty dstbuton functon of a sum of ndeendent andom vaables and ts alcaton to the sum of Raylegh andom vaables, IEEE Tans Commun, vol 38, no 9, , Set 99 [9] M Smon and M-S Aloun, A unfed aoach to efomance analyss of dgtal communcaton ove genealzed fadng channels, n Poc IEEE, vol 86, Set [] M-S Aloun and M Smon, Pefomance analyss of coheent equal gan combnng ove Naagam-m fadng channels, Submtted fo ublcaton n IEEE Tans Veh Technol [] Kaagannds and S Kotsooulos, Exact evaluaton of egual-gan dvesty n the esence of Naagam fadng, IEE Electon Lett, vol 36, no 4, July [] Y-C Ko, M-S Aloun, and M Smon, Outage obablty of dvesty systems ove genealzed fadng channels, n Poc IEEE Int Sym Infom Theoy ISIT ), Soento, Italy, June [3] M-S Aloun and A oldsmth, A unfed aoach fo calculatng eo ates of lnealy modulated sgnals ove genealzed fadng channels, IEEE Tans Commun, vol 47, no 9, , Set 999 [4] M Smon and M-S Aloun, Dgtal Communcatons ove Fadng Channels: A Unfed Aoach to Pefomance Analyss, st ed, Wley & Sons, July [5] A Abu-Dayya and N Beauleu, Mcodvesty on Rcan fadng channels, IEEE Tans Commun, vol 4, no 6, 58 67, June 994 [6] A Abu-Dayya and N Beauleu, Dvesty MPSK eceves n cochannel ntefeence, IEEE Tans Veh Technol, vol 48, no 6, , Nov 999 [7] S Kotsooulos and Kaagannds, Eo efomance fo equal-gan combnes ove Raylegh fadng channels, IEE Electon Lett, vol 36, no, May [8] bson, The Moble Communcatons Handboo, CRC and IEEE Pess, 996 [9] S Kotsooulos and Kaagannds, Moble Communcatons, ee ed, Paasotou SA Publcaton, 997 [] A Paouls, Pobablty, Random Vaables, and Stochastc Pocesses, 3d ed, Mcaw-Hll, 99 [] M Abamovtz and I Stegun, Handboo of Mathematcal Functons wth Fomulas, ahs, and Mathematcal Tables, 9th ed, New Yo, NY: Dove Publcatons, 97 [] M Naagam, The m-dstbuton A geneal fomula of ntensty dstbuton of ad fadng, n Statstcal Methods n Rado Wave Poagaton, 3 36, Pegamon Pess, Oxfod, UK, 96 [3] A Annamala, C Tellambua, and V Bhagava, Exact evaluaton of maxmal-ato and equal-gan dvesty eceves fo M-ay QAM on Naagam fadng channels, IEEE Tans Commun, vol 47, no 9, , Set 999 [4] C Tellambua, Cochannel ntefeence comutaton fo abtay Naagam fadng, IEEE Tans Veh Technol, vol 48, no, , Ma 999 [5] L Wang and C Lea, Co-channel ntefeence analyss of shadowed Rcan channels, IEEE Commun Lett, vol, no 3, 67 69, Ma 999 eoge K Kaagannds was bon n Pthagoon Samos, eece n 963 He eceved ts dloma n 987 and the PhD degee n 999, both n electcal engneeng, fom the Unvesty of Patas, Patas, eece Fom 99 to 993, hs eseach focused on the develoment of ntefaces fo dffuse IR communcatons lns Snce 994, he has been develong otmum channel assgnment schemes and methods fo the movement of QoS and os n moble ado systems Pesently, he wos as a Reseach Fellow n Insttute of Sace Alcatons & Remote Sensng, Natonal Obsevatoy of Athens and as a Vsto Pofesso n Technologcal Insttute of Lama, eece Hs eseach nteests nclude communcatons theoy, dgtal communcatons ove fadng channels, moble ado systems, ntefeence oblems, and QoS n weless netwos D aagannds has ublshed and esented about 35 techncal aes n scentfc jounals and oceedngs of confeences and he s co-autho n a ee Edton Boo on Moble Communcatons He s evewe fo ntenatonal jounals as the IEEE Communcatons Lettes, IEE Poceedng-Communcatons, etc D aagannds s a membe of Techncal Chambe of eece, a membe of IEEE, a membe of the IEEE ComSoc, a membe of the IEEE Infomaton Theoy Socety, and a membe of the IEEE Vehcula Technology Socety Stavos A Kotsooulos was bon n Agos-Agoldos eece) n the yea 95 He eceved hs BSc n Physcs n the yea 975 fom the Unvesty of Thessalon, and n the yea 984 got hs Dloma n Electcal and Comute Engneeng fom the Unvesty of Patas He s an MPhl and PhD holde snce 978 and 985 coesondngly He dd hs ostgaduate studes n the Unvesty of Badfod n Unted Kngdom Cuently he s membe of the academc staff of the Deatment of Electcal and Comute Engneeng of the Unvesty of Patas and holds the oston of Assstant Pofesso He develos hs ofessonal lfe teachng and dong eseach at the Laboatoy of Weless Telecommuncatons Unv of Patas), wth nteest n moble communcatons, ntefeence, satellte communcatons, telematcs, communcaton sevces and antennae desgn Moeove he s the co-autho of the boo ttled moble telehony The eseach actvty s documented by moe than 6 ublcatons n scentfc jounals and oceedngs of confeences Ast Pofesso Kotsooulos has been the leade of seveal ntenatonal and many natonal eseach ojects Fnally, he s membe of the ee Physcsts Socety and membe of the Techncal Chambe of eece