IN many recent wireless communication systems, both licensed,

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1 74 IEEE TRASACTIOS O COMMUICATIOS, VO. 63, O. 3, MARCH 15 Moble Communcaton Systems n the Presence of Fadng/Shadowng, ose and Interference Petros S. Bthas, Member, IEEE, and Athanasos A. Rontoganns, Member, IEEE Abstract In ths paper, the effects of nterference on composte fadng envronments, where multpath fadng coexsts wth shadowng, are nvestgated. Based on some mathematcal convenent expressons for the sum of squared K -dstrbuted random varables, whch are derved for the frst tme, mportant statstcal metrcs for the sgnal to nterference and nose rato SIR are studed for varous cases ncludng non dentcal, dentcal and fully correlated statstcs. Furthermore, our analyss s extended to mult-channel recevers and n partcular to selecton dversty SD recevers, nvestgatng two dstnct cases, namely, sgnal-tonose rato based and SIR-based SD. For all scenaros, smplfed expressons are also provded for the nterference lmted case, where the nfluence of thermal nose s gnored. The derved expressons are used to analyze the performance, n terms of the average bt error probablty ABEP and the outage probablty OP, of such systems operatng over composte fadng envronments. A hgh SR analyss s also presented for the OP and the ABEP, gvng a clear physcal nsght of the system s performance n terms of the dversty and codng gans. The analyss s accompaned by numercal evaluated results, clearly demonstratng the usefulness of the proposed theoretcal framewor. Index Terms Composte fadng channels, mnmum number of antennas, selecton dversty, sgnal-to-nterference and nose rato, sum of squared K -dstrbuted RVs. I. ITRODUCTIO I many recent wreless communcaton systems, both lcensed, e.g., long term evoluton TE, or unlcensed, e.g., WF or bluetooth, a user qute often shares the same channels wth other users and thus n the recever sde the sgnals need to be ntellgently separated. Ths s mperatve, snce the aggressve frequency reuse that s frequently employed n cellular systems for ncreasng spectrum effcency, causes co-channel as well as adjacent channel nterference. The cochannel and adjacent channel nterference depend on varous physcal factors, ncludng nterferers spatal dstrbuton, nterferng channels fadng, the power of the nterferers and the wreless communcaton system consdered. Dependng upon Manuscrpt receved June 17, 14; revsed October 3, 14 and ovember 8, 14; accepted December 8, 14. Date of publcaton January 1, 15; date of current verson March 13, 15. Ths research has been co-fnanced by the European Unon and Gree natonal funds through the Operatonal Program Educaton and felong earnng of the atonal Strategc Reference Framewor SRF-Research Fundng Program: Thales. Investng n nowledge socety through the European Socal Fund. A prelmnary verson of a part of ths wor has presented at IEEE Symposum on Sgnal Processng and Informaton Technology ISSPIT, Athens, Greece, Dec. 13. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was Y. Chen. The authors are wth the Insttute for Astronomy, Astrophyscs, Space Applcatons and Remote Sensng of the atonal Observatory of Athens, Pentel, 1536, Greece e-mal: pbthas@space.noa.gr; tronto@noa.gr. Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer 1.119/TCOMM the fadng characterstcs as well as the exstence or not of multchannel transmtters/recevers and/or relays, numerous contrbutons analyzng the effects of fadng n conjuncton wth nterference have been made, e.g., [] [7] and the references theren. In terrestral ndoor or outdoor and satellte land-moble systems, the ln qualty s also affected by slow varatons of the mean sgnal level due to the shadowng from terran, buldngs and trees [8]. Under these crcumstances, where multpath fadng coexsts wth shadowng, the so-called composte fadng/ shadowng envronment orgnates. In the past, ths envronment has been statstcally modeled by usng lognormal-based dstrbutons such as Raylegh-, aagam- and Rce-lognormal [8] [1] and therefore rather mathematcally complcated expressons have been derved for the performance analyss on such communcaton scenaros. To facltate the communcaton systems performance evaluaton n these envronments, new famles of dstrbutons that accurately model composte fadng condtons have been proposed, most notably as the K and the generalzed-k K G dstrbutons, e.g., [11], [1]. Based on the mathematcal tractablty of these new composte fadng models, research efforts have been made recently for nvestgatng the nfluence that nterference has to the system s performance, e.g., [1], [13] [15]. A common practce n all these wors s that the research was restrcted to nterference-lmted wreless communcaton systems and thus only the statstcs of the sgnal-to-nterference-rato SIR was studed. In ths paper extendng ths approach, and n order to provde a more complete stochastc analyss framewor of the composte fadng envronment, the effect of thermal nose s also taen nto consderaton. Ths s mportant snce thermal nose may be the man source of system performance degradaton especally n cases of wea nterferng sgnals. Therefore, assumng such a complete model, our contrbuton n ths paper can be summarzed as follows: mathematcal convenent expressons for the sum of squared K -dstrbuted random varables RVs are derved for the frst tme; based on these expressons, the sgnal to nterference and nose rato SIR statstcs are studed for varous scenaros ncludng non dentcal, dentcal and fully correlated fadng/shadowng effects; the analyss s extended to sngle-nput multple-output SIMO recevers and n partcular to selecton dversty SD recevers that operate n such envronments; the derved expressons are used to nvestgate the system performance n terms of the average bt error probablty ABEP and the outage probablty OP; IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 BITHAS AD ROTOGIAIS: MOBIE SYSTEMS I FADIG/SHADOWIG, OISE AD ITERFERECE 75 Fg. 1. System model. a hgh SR analyss s also provded for the ABEP and the OP n the SIMO case, whch s then employed to dentfy the codng and dversty gans of the system under study. For all scenaros studed, smplfed expressons are also provded for the nterference lmted cases, where the nfluence of thermal nose s gnored, whlst for SIMO dversty systems an nvestgaton on the power effcency s also presented. The remander of ths paper s organzed as follows. The system model s descrbed n Secton II. In Secton III the statstcs of the sum of squared K -dstrbuted RVs are nvestgated for the three scenaros under consderaton, namely non dentcal, dentcal and fully correlated statstcs. The statstcs of the SIR for sngle-nput sngle-output SISO and SIMO channels are derved n Sectons IV and V, respectvely. In Secton VI the performance analyss of such systems s presented, whle performance evaluaton results are provded n Secton VII. Fnally, concludng remars are gven n Secton VIII. II. SYSTEM MODE We consder a wreless-moble communcaton system, wth a sngle-antenna transmtter and n general a multpleantennas recever, where each recever dversty branch s experencng nterference comng from sources, as shown n Fg. 1. In a typcal macrocellular moble rado networ, as t s the one consdered n ths study, the moble users are usually surrounded by local scatterers so that the plane waves arrve from many drectons wthout a drect lne-of-sght os component [16]. For such a type of scatterng envronment the receved envelope s Raylegh dstrbuted. Ths assumpton holds for both the desred and nterferng sgnals. However, n many wreless networs, due to the expected larger propagaton dstances, nterferng sgnals are very lely to propagate over obstructed paths, and thus experence, n addton, severe shadowng condtons. Ths does not hold for the desred sgnals, snce not many obstacles between the user and the tagged base staton are expected to be present due to the comparatvely small dstance between them. It s noted that smlar assumptons concernng the nvolved desred and nterferng channel models n varous settngs have been made by many other researchers n the past, e.g., [16, pp. 14], [17] [1]. It s also mportant to note that the adopted channel model offers also ncreased tractablty, and thus gve us the opportunty to analytcally nvestgate the nfluence of the combnaton of fadng, shadowng and multchannel recepton to the system performance. We also assume that n general the level of nterference at the recever s such that the effect of thermal nose on system performance cannot be gnored []. Havng defned our model, the complex baseband sgnal y n receved at the nth antenna, can be expressed as y n = h Dn s D + h In, s I + w n 1 where h Dn represents the complex channel gan between the transmtter and the nth recever antenna wth ts envelope followng the Raylegh dstrbuton and s D s the desred transmtted complex symbol wth energy E sd = E s D, and E denotng statstcal averagng. Furthermore, n 1, h In, represents the complex channel gan wth ts envelope followng the K -dstrbuton of the nterferng sgnal s I and w n s the complex addtve whte Gaussan nose AWG wth zero mean and varance. To proceed, we denote the nstantaneous sgnal-tonose-rato SR of the desred sgnal as Dn = h Dn E sd /, the correspondng average SR as Dn = E h Dn E sd /, whlst the nstantaneous nterference-to-nose-rato IR of the th nterferng sgnal s defned as n, = h In, E si /, wth correspondng average IR equal to n, = E h In, E si /. Snce the desred sgnal s subject only to multpath fadng, ts nstantaneous SR Dn at the nth antenna can be assumed to be exponentally dstrbuted. Specfcally, consderng ndependent fadng condtons, the probablty densty functon PDF of Dn s gven by f Dn x= 1 exp x. Dn Dn Here, the nstantaneous IR, n,, of the nterferng sgnals, whch are subject to fadng/shadowng effects, s assumed to follow a squared K dstrbuton 1 wth PDF gven by [11] f In, x= n, n, n, +1 x n, 1 Γ n, K n, 1 n, x. 3 In 3, n, denotes the shapng parameter of the dstrbuton, Γz = exp ttz 1 dt [, eq. 8.31/1] and K v s the second nd modfed Bessel functon of vth order [, eq. 8.47/1]. In addton, n, s related to the severty of the shadowng, e.g., for small values for n,, 3 models severe shadowng condtons, whle as n,, t approxmates the exponental dstrbuton. The correspondng output SIR, at the nth recever antenna, s expressed as [] outn = D n n where n denotes the total IR,.e., n = n,, and the PDF of n, s gven by 3. The PDF of outn can be evaluated as follows f outn = n, 1 + x f Dn [1 + x] f In xdx. 5 1 For smplfcaton purposes and to avod repettons, from now on squared K dstrbuton wll be referred as K. Wthout loosng the generalty, t s assumed that all dversty branches are affected by the same nterferng sources.

3 76 IEEE TRASACTIOS O COMMUICATIOS, VO. 63, O. 3, MARCH 15 In the next secton mportant statstcal metrcs of n wll be evaluated consderng non dentcal, dentcal as well as fully correlated statstcal parameters. The derved results presented n Secton III, wll be used to obtan expressons for the nstantaneous output SIR of both SISO and SIMO systems presented n Sectons IV and V, respectvely. TABE I MIIMUM UMBER OF TERMS H OF 7 REQUIRED FOR OBTAIIG ACCURACY BETTER THA ±1 5 III. O THESUM OF K -DISTRIBUTED RVs In ths secton the statstcs of the sum of K -dstrbuted RVs wll be nvestgated for three dfferent scenaros, namely for K -dstrbuted RVs whch, are ndependent but non dentcally dstrbuted.n.d., are ndependent and dentcally dstrbuted..d. and have fully correlated mean values. It should be noted here that n the past, several efforts have been devoted to accurately approxmatng the K -dstrbuton, e.g., [3]. However, to the best of the authors nowledge, none of them led to exact expressons for the sum of K RVs, whch are obtaned for the frst tme n the followng analyss. Snce the analytcal results provded n ths and the next sectons refer to the arbtrary nth antenna, the antenna ndex n wll be omtted and wll be reestablshed n Secton V. A. Independent But ot Identcally Dstrbuted RVs Theorem 1: et denote a RV defned as Δ = 6 where the PDF of s gven by 3. Consderng non dentcal nterference statstcs, the PDF of can be expressed as f I =G 1 +, λ 1,...,λ In 7, S z x q,y q = + j S, h= S,1 1 S, I S S,1 1,I G 1 G 3 I G 1 S,I. 7 } {{ } IS z x q q=1 [ y q, G 1 = Γ1 ], G j = j λ + h, G 3 = t h,λ1, f = 1, G 3 = t 1,h = h t p,λ1 t h p,λ,f =, G 3 = t 1,h = h p= p= +h t,p t h p,λ,f >, wth t h,λ1 = 1h λ1 /λ1 1 λ 1 h!γ1 λ1 1 λ1 +h and I v denotng the frst nd modfed Bessel functon of vth order [, eq. 8.46/1]. Proof: See Appendx A. It can be shown that the nfnte seres n IS converges everywhere. 3 In addton the rate of convergence of f I gven n 7, was nvestgated expermentally. Ths s mportant because n practce, to evaluate f I, the nfnte seres n IS s truncated and a number of seres terms H s retaned. In Table I the mnmum values of H, requred for accuracy better than ±1 5 are presented for varous values of the average IR and dstrbuton s parameters. These results have been obtaned by assumng an exponentally decayng profle,.e., = 1 db/1 exp[ d 1], = 3.3 wth = 1,,...,, d =.1 and = 3. It s clearly depcted from ths table that a relatvely small number of terms s suffcent to acheve a hgh accuracy and as a consequence the PDF n 7 converges fast. ote that H manly depends on the value of the varable of the Bessel functon I v n 7, and ncreases as ths varable ncreases. As a result, H ncreases as decreases and/or, ncrease. Moreover, comparng these results wth other employng nfnte seres, e.g., [5], [6], the requred number of terms s sgnfcantly smaller for a smlar target accuracy. emma 1: For.n.d. statstcs, the CDF of can be obtaned as S F I =G 1, I S S,1,I + S,, h= λ 1,...,λ S, G G 3 I G S,I. 8 Proof: Substtutng 7 n the defnton of the CDF, F I = f xdx, mang changes of varables of the form x 1/ = y and y = z and usng [, eq /7],.e., 1 xv+1 I v axdx = a 1 I v+1 a, 8 s obtaned. In Fg. a, several PDFs, obtaned by usng 7, are plotted for varous values of the number of nterferers. These dstrbutons have been evaluated by also assumng exponental decayng average IR,.e., = 1 db/1 exp[ d 1], =.3 wth = 1,,...,, and db = 15, d =.1. Addtonally, smulated results are also ncluded n ths fgure, verfyng n all cases the agreement between the analytcal and the smulated PDFs. 3 Detaled analyss s avalable n the techncal report [4], whch has been also prepared.

4 BITHAS AD ROTOGIAIS: MOBIE SYSTEMS I FADIG/SHADOWIG, OISE AD ITERFERECE 77 emma : For..d. statstcs, the CDF of obtaned as F I = = Γ1 1 c h h= can be G 4 G 4 I G Proof: Substtutng 9 n the defnton of the CDF, F I = f xdx, mang changes of varables of the form x 1/ = y and y = z and usng agan [, eq /7], 1 s obtaned. Fg.. a The PDF of gven n 7. b The PDF of out gven n 17. B. Independent and Identcally Dstrbuted RVs Theorem : For the same RV defned n 6, consderng dentcal nterference statstcs,.e., = and =, the PDF of can be expressed as f I = = Γ1 1 c h h= 1+G 4 G 4 1 I G where c = a, c h = ha 1 [ ] h t=1 t h +ta t c h t for h 1a = 1 1, a t = 1t t!1 +t,wth G 4 = + h +1. Proof: See Appendx B. C. Fully Correlated Shadowng When the dfferent nterferng paths exhbt dentcal shadowng effects, the mean values of the correspondng K -dstrbuted RVs are fully correlated. Ths s the so-called fully or totally correlated shadowng communcaton scenaro, whch has ganed consderable nterest lately, e.g., [7] [3], and wll be nvestgated n ths secton. Ths type of shadowng arses n stuatons where the nterferers have approxmately the same dstance from the recever, and thus the same obstacles shadow n a qute smlar way the varous nterferng sgnals. As a result, the local mean powers of the nterferng sgnals become correlated [14], [31], a stuaton that qute often arses n ndoor communcaton systems [3]. As t wll be shown below, the fully correlated shadowng communcaton scenaro s mathematcally tractable compared to the more general models descrbed above, n the sense that easy-to-compute closed-form expressons can be easly derved, whch provde useful nsghts to the system performance. In such a communcaton scenaro, where the multpath Raylegh components of the nterferers are ndependent but all of them experence a common local average power,, t has been shown that the moments generatng functon MGF of can be expressed as [11, eq. 9] M I s= U, + 1, 11 s s where U denotes the confluent hypergeometrc functon defned as U, + 1, I s = Γ Γ 1F 1 ; + 1; I s + Γ Γ I s 1F 1 ;1 + ; I s [, eq. 9.1/], wth 1F 1 α,,z = α z =! representng the confluent hypergeometrc functon [, eq. 9.1/1] and v denotng the pochhammer symbol [, pp. xl]. Based on ths expresson and the defnton of 1 F 1, 11 yelds M I s= Γ Γ s = + Γ Γ [/ s]! + 1 s = [/ s]!1 +. 1

5 78 IEEE TRASACTIOS O COMMUICATIOS, VO. 63, O. 3, MARCH 15 Applyng the nverse aplace transform n 1, yelds the followng expresson for the PDF of f I = Γ ΓΓ 1 = Γ + ΓΓ /! = /. 13!1 + Furthermore, based on the defnton of the generalzed hypergeometrc functon,.e., F 1 b;z = z /!b = [33, eq ], and by employng [33, eq ], a smpler closed-form expresson for the PDF of, when fully correlated shadowng effects are present, can be extracted as f I = / K. 14 ΓΓ ote that n a dfferent research topc,.e., free space optcal communcatons, a smlar PDF expresson, as the one gven n 14, has been ndependently derved n [34]. Substtutng now 14 n the defnton of the CDF [35, eq. 4.17], usng the Mejer-G functon representaton for the K [33, eq ],.e., K v z = 1 G, z v, 4, v, and then [36, eq. 6], the CDF of can be expressed n a smple closed form as F I = / + ΓΓ + G,1 1,3 1 +,, + 15 where G m,n p,q [ ] s the Mejer s G-functon [, eq. 9.31]. Obvously, the derved expressons for the fully correlated case of shadowng, can be drectly employed for further analytcal purposes. Moreover, as t wll be shown n Secton VII, the system performance results, whch are obtaned for the fully correlated case, are qute close to those obtaned for the..d. case. Fnally, t should be noted that the prevously derved expressons for the sum of K -dstrbuted RVs can also be appled to analyze the performance of varous recever structures, ncludng a maxmal rato combner output SR study. IV. SIR STATISTICS FOR SISO SYSTEMS In ths secton, based on the prevously derved expressons for the sum of K -dstrbuted RVs, a statstcal analyss of the nstantaneous output SIR out s presented for the three communcaton scenaros under consderaton, namely.n.d.,..d. and fully correlated nterference condtons. In all cases smplfed expressons for the SIR are also obtaned. A. on Identcal Interference Statstcs Substtutng the PDF for the SR gven by and the PDF for the IR gven by 7 n 5, t can be shown that ntegrals of the followng form appear I 1 = Φ q 1, dx 16 [ where Φa,b =x a xexp 1+x ]I a 1 bx wth D q 1 = or q 1 = G h. Ths type of ntegrals can be solved n closed form usng [, eq /]. After some straghtforward mathematcal manpulatons the PDF of out under.n.d. nterference condtons can be expressed as f out = exp / D D G 1 exp S, / D S D,1 M S, S, 1, S,1 1 / D S, + +M S,, S,1 1 D S, I D G, λ 1,...,λ G D M G 1, G 1 +M G, G 1 G 3 h= S, / D D S, S, / D S, 17 where M λ,µ s the Whttaer functon [, eq. 9./], whch s a bult-n functon n many mathematcal software pacages. Here, t should be mentoned that 17 represents a vald PDF, snce t s a nonnegatve functon, and usng [33, eqs and ] and [36, eq. 1], t can be verfed that f out d = 1 [37]. In Fg. b, several PDFs, obtaned by usng 17, are plotted for the same parameters as the ones used n Fg. a and varous values for the number of nterferers. The smulated results also ncluded n ths fgure depct n all cases the tght agreement between analytcal and smulated PDFs. Furthermore, for the nfnte seres appearng n 17, a smlar rate of convergence has been observed wth that of the seres appearng n 7. By defnton, the CDF of the nstantaneous output SIR s expressed as F out = F D [1 + x] f I xdx. 18 Substtutng the exponental CDF,.e., F D x=1 exp x D 19 and 7 n 18, followng a smlar procedure as the one used for dervng 17, yelds the followng closed-form expresson

6 BITHAS AD ROTOGIAIS: MOBIE SYSTEMS I FADIG/SHADOWIG, OISE AD ITERFERECE 79 TABE II SISO SIR STATISTICS for the CDF of out 4 F out =1 exp S, D G 3, h= λ 1,...,λ + S, G 1 exp S, D / D S, M S,, S,1 1 D S, I D G M G, G 1 S,. / D For the specal case of = 1K -dstrbuted nterferer, smplfes to F out =1 exp D D I D D exp W, 1. 1 In a smlar communcaton scenaro consstng of a aagam-m desred sgnal and a aagam-lognormal nterferng sgnal, an ntegral expresson for the outage performance was derved n [16, eq. 3.59]. Settng n that expresson m = 1,.e., consderng Raylegh/Raylegh-lognormal fadng/shadowng condtons, and comparng the resultng formula wth 1, the mathematcal smplfcaton offered by the latter s obvous. 4 A smpler expresson for ths CDF, can be obtaned usng the MGF-based approach that s proposed n [3]. Smplfed Expressons for the SIR: In many cases, the moble communcaton systems tend to be nterference lmted rather than nose lmted, snce the thermal and man-made nose effects are often nsgnfcant compared to the sgnal levels of cochannel users [38]. Ths case wll be also studed here, where consderng an nterference lmted envronment,.e., gnorng the AWG at the user termnal, the receved SIR s gven by out = D whlst ts PDF and CDF expressons are f out = F out = xf D x f I xdx F D x f I xdx 3 respectvely. Substtutng and 7 or 19 and 7 n 3, and followng a smlar procedure as the one for dervng 17, yelds the PDF and CDF expressons, respectvely, for the.n.d. case gven n Table II. B. Identcal Interference Statstcs Substtutng and 9 n 5 and after some mathematcal procedure yelds f out = Γ1 1 D n n=c = G 4 [ Φ G 4, ] dx. 4

7 73 IEEE TRASACTIOS O COMMUICATIOS, VO. 63, O. 3, MARCH 15 The ntegral n 4 can be solved n closed form usng [, eq /] and thus the PDF of out under..d. nterference condtons can be expressed as f out = exp / D D = D = Γ1 1 G 4 [ D D D c n exp M G n= 4 I, G G ] 4 D D M G 4 +, G The correspondng expresson for the CDF s gven by F out =1 exp Γ1 1 n= G 4 D c n exp D M G 4, G 4 1 D. 6 For the SIR case smplfed expresson for the PDF and CDF are gven n Table II. C. Fully Correlated Interference Statstcs Substtutng and 14 n 5 yelds the followng expresson for the SIR of f out + f out = / D ΓΓ [ exp x x } {{ } I ]K 1 + x D x 1/ dx. 7 } {{ } I After performng some straghtforward mathematcal manpulatons and usng [, eq /3] a closed-form expresson for the PDF of out can be derved as f out = D exp D exp D D [ D W1, + ] D D W ++1, 8 where W λ,µ s the Whttaer functon [, eq. 9./4]. Substtutng the exponental CDF and 14 n 18 and usng [, eq /16],.e., xµ K v αxdx = µ 1 Γ α µ+1 Γ 1+µ+v 1+µ v and [, eq /3], yelds the followng closed-form expresson for the CDF of out + 1 D F out =1 exp D exp D W1, D. 9 Smplfed Expressons for the SIR: The closed-form expresson for the PDF and the CDF of out are gven n Table II. V. SIR STATISTICS FOR SIMO: THE FUY CORREATED CASE In ths secton, focusng on the fully correlated shadowng case for the nterferng sgnals, we consder a SD recever and nvestgate two dstnct selecton technques, namely the SRbased and the SIR-based. We also assume that the receve antennas are suffcently spaced, so that the nterferng sgnals receved n any of them are totally ndependent from the ones receved by any other antenna. In ths context, new closed-form expressons are derved for mportant statstcal metrcs of the nstantaneous output SIR of the two technques under consderaton. It s noted that the analytcal framewor presented here can also be appled to the.n.d. as well as to..d. nterference scenaros. However, due to space lmtatons these results are not presented here. In [1], some prelmnary results lmted to the SR-based SD scenaro were presented. A. SR-Based Crteron For the SR-based SD crteron n the presence of AWG and multple nterferng sgnals, the dversty recever montors the avalable dversty branches contnuously and selects the branch wth the largest nstantaneous SR for data detecton. Ths SD technque requres the separaton of the desred sgnal from the nterferng sgnals, whch can be practcally acheved by usng dfferent plot sgnals for each of them [4]. Mathematcally speang, the nstantaneous system output SIR of ths SIMO system can be expressed as SDout = SD where SD = max{ D1, D,..., D } represents the nstantaneous output SR of the SD recever, wth Dn denotng the SR of the nth branch, followng the PDF gven by. The CDF of SD for.n.d. fadng condtons, 5 s gven by F SD x= n=1 F Dn x. 31 For..d. fadng condtons, 31 smplfes to F SD x = [F D x], wth F D x gven n 19. Based on A.4 and after some straghtforward mathematcal manpulatons, F SD x can be expressed as F SD x=1 + exp S1, n x n,n 3 λ 1,...,λn whch for the case of..d. fadng condtons smplfes to F SD x= 1 n exp n x. 33 n D n= 5 For SR-based SIMO, the.n.d. and..d. condtons refer to the fadng statstcs of the nstantaneous SRs of the desred sgnal at the branches of the recever. For the nterferers, fully correlated shadowng has been assumed.

8 BITHAS AD ROTOGIAIS: MOBIE SYSTEMS I FADIG/SHADOWIG, OISE AD ITERFERECE 731 TABE III SIMO SIR STATISTICS Startng from 3 and substtutng 3 and 14 n 18, ntegrals of the form I appearng n 7, are obtaned. Therefore followng the procedure proposed n Secton IV-C, the CDF of SDout, can be obtaned n closed form as F SD out =1+ n,n λ 1,...,λn / S1, n exp / S1, n n,n λ 1,...,λn + 1 W1, exp S1, n /. S1, n 34 Its correspondng PDF s gven by f = SD S n1, exp S n out 1, / S1, n W1, + 1 exp / S1, n / S1, n + S1, n W / ++1,. S1, n 35 Consderng..d. fadng condtons 34 smplfes to F =1 SD out 1 n+1 exp n n=1 n D + 1 D D D exp W1 n n, n 36 whlst 35 smplfes to f SD out = n=1 1 n n exp n n D D + 1 { D D D exp W1 n n, n n 1 } D + W 1 D,. 37 n Smplfed Expressons for the SIR: For the SIR case, the nstantaneous system output SIR can be expressed as SDout = SD. 38 The smplfed PDF and CDF expressons for both.n.d. and..d. cases are gven n Table III. B. SIR-Based Crteron Under a SIR-based crteron, the dversty recever selects the branch wth the hghest nstantaneous SIR for coherent detecton. Ths technque s more complex to be mplemented, than the SR-based one, snce computatonally demandng processng operatons are requred at the recevng end [4]. These nclude sgnal separaton, SIR calculaton per branch, statstcal orderng of the resultng SIRs and SIR-based selecton va a maxmum selecton crteron. For the SIRbased scenaro the nstantaneous SIR of the system output can be expressed as SDout = max out1, out,..., out, where outn represents the nstantaneous SIR of the nth branch, wth PDF and CDF gven by 8 and 9, respectvely. Therefore,

9 73 IEEE TRASACTIOS O COMMUICATIOS, VO. 63, O. 3, MARCH 15 consderng.n.d. fadng condtons 6 the CDF of the output SIR can be expressed as [ ] F = SD out outn. 39 n=1f For the..d. fadng case, 39 smplfes to F SD out =[ F out ]. 4 The correspondng expressons for the PDFs, consderng.n.d. fadng condtons, are f = SD out f outn n=1 m=1 m n For..d. fadng condtons 41 smplfes to F outm. 41 f SD out =f out [ F out ] 1. 4 Smplfed Expressons for the SIR: The CDF and PDF for the.n.d. fadng case can be obtaned by substtutng the PDF and the CDF expressons of the fully correlated case presented n Table III n 39 and 41, respectvely. For..d. fadng, the same expressons should be substtuted n 4 and n 4. VI. PERFORMACE AAYSIS: THE FUY CORREATED CASE In ths secton, focusng also on the fully correlated case and usng the prevously derved expressons for the nstantaneous output SIR of the SD recevers under consderaton, mportant performance qualty ndcators are studed. In partcular, the performance of both SR-based and SIR-based SD recevers s evaluated usng the OP and the ABEP crtera. In addton, n all cases an asymptotc analyss s ncluded for the hgh SR regme. A. Outage Probablty OP The OP s defned as the probablty that the SIR falls below a predetermned threshold th and s gven by P out = F out th. 1 SR-Based Crteron: Consderng the SR-based crteron, the OP can be obtaned by usng 34 for.n.d. fadng or 36 for..d. fadng. The correspondng CDF expresson for the SIR case can be found n Table III. Hgh SR Approxmaton: The exact results presented n the prevous sectons do not provde a clear physcal nsght of the system s performance. Therefore, here, we focus on the hgh SR regme to obtan mportant system-desgn parameters such as the dversty gan G d and the codng gan G c. Addtonally, these analytc expressons help to quantfy the amount of performance varatons, whch are due to the nterferng 6 For the SIR-based crteron the.n.d. and..d. condtons refer to the fadng statstcs of the nstantaneous SIRs outn. effects as well as to the recever s archtecture. At hgh SR the exponental CDF can be closely approxmated by F D x x D [39]. Based on ths approxmated expresson, and assumng..d. fadng condtons the.n.d. case can be smlarly analyzed the CDF of SD can be expressed as F SD x=[f D x]. Therefore followng the procedure proposed n Secton IV-C, and employng [, eq /16], the OP of SDout, for hgh SR, can be expressed as F SD out th [ ] Γ + nγ + n n= n / n th D ΓΓ. 43 }{{} D 1 It s obvous that 43 s of the form G c D G d, where G d represents the dversty gan, whch as expected equals, and G c = D 1/ 1 s the codng gan [4], [41]. Therefore, the codng gan of the system s affected by the number of branches, the average IR, the outage threshold th, the severty of the shadowng effects on the nterferng channels, descrbed by, and the number of nterferers, wth the frst three parameters havng the hghest mpact. In partcular, the codng gan ncreases as and/or and/or and/or th ncrease or decreases. As far as and th are concerned, t s obvous that ther ncrease wll result to a hgher codng gan. Regardng the shadowng condtons on the nterferng sgnals, t can be shown from 43 that as they get worst,.e., decreases or ncreases, the codng gan ncreases. Ths s a reasonable result snce worst channel condtons on the nterferng sgnals lead to an ncreased SIR and thus to an mproved codng gan. SIR-Based Crteron: Consderng the SIR-based crteron, the OP can be obtaned by substtutng 9 n 39 for.n.d. fadng or 4 for..d. fadng. For the SIR case, the correspondng CDF expresson, whch should be substtuted 39 for.n.d. fadng or 4 for..d. fadng, can be found n Table II. Hgh SR Approxmaton: For hgh values of the average SR, by followng a smlar procedure as the one used for the SR-based case, the OP can be closely approxmated as F SD out th 1 + th D 44 }{{} D where G d =, G c = D 1/, whle smlar conclusons wth the SR-based case can be drawn for the dversty and codng gans. On the other hand, as t s also noted n [4], G c s ndependent of the nterferng sgnals parameter. Ths does not apply to the SR-based case due to ts dfferent mode of operaton. B. Average Bt Error Probablty ABEP The ABEP wll be evaluated by usng the MGF and CDF based approaches as descrbed next. 1 SR-Based Crteron: Consderng.n.d. fadng condtons, substtutng 35 n the defnton of the MGF, usng

10 BITHAS AD ROTOGIAIS: MOBIE SYSTEMS I FADIG/SHADOWIG, OISE AD ITERFERECE 733 [33, eq ], and employng [, eq /1], yelds the followng expresson for the MGF of SDout S n1, M SD out s= n,n λ 1,...,λn ΓΓ S n 1, +s S n 1, S 1, n 1 I S1, n + s G 1,3 3,1 S1, n +s S 1, n 1 I + S1, n G 1,3 3,1 S1, n + s , 1 +, , 1 +, For the..d. fadng case, based on 37, 45 smplfes to M s= SD out n=1 1+ s 1 D { n G 1,3 ni / 3,1 ΓΓ n+s D + 1+ s D G 1,3 ni / 3,1 n n + s D whch for the SISO system becomes M out s= D / + ΓΓ D [ ] n+1 n/d + s D n n + 1, +1, , +1, { s D } G 1,3 I / D + +,1,1+ 1 3,1 1/ D + s + + +s D D G 1,3 I / D + } + 1,1,1 + 3,1 1/ D + s For the SIR and for both.n.d. and..d. fadng condtons, MGF expressons for SDout are ncluded n Table III or Table II for the SISO system. Usng the prevously derved MGF expressons and followng the MGF-based approach, the ABEP can be readly evaluated for a varety of modulaton schemes [8]. More specfcally, the ABEP can be calculated: drectly for non-coherent dfferental bnary phase shft eyng DBPSK, that s Pb DBPSK =.5M SD 1; and va numercal ntegraton for Gray encoded M-PSK, that s P out M PSK 1 πlog M π π/m [ log M sn M π/m SD out sn φ ] dφ. b = Hgh SR Approxmaton: To evaluate the ABEP performance at the hgh SR regme, the CDF-based approach wll be employed. Specfcally, the ABEP can be evaluated as [43] P b = P ef SD d. 48 out where P e denotes the negatve dervatve of the condtonal error probablty. For example assumng DBPSK modulaton P e =αβexp β, where α = 1/,β = 1[43]. Substtutng 43 n 48 and usng [, eq /3],.e., xn exp µxdx = n! µ n+1, yelds the followng approxmaton for the ABEP P b [ n= ] αγ + nγ + nγ + 1 n / n ΓΓβ D 49 }{{} D 3 and G d =,G c = D 1/ 3. It should be noted that smlar observatons as the ones obtaned for 43 can be also made for 49. SIR-Based Crteron: To evaluate the ABEP performance for the SIR-based crteron the CDF-based approach s used. For the.n.d. fadng case, 39 s substtuted n 48, whlst for..d. fadng 4 s employed. For both cases numercal ntegraton technques must be appled, usng any of the well-nown mathematcal software pacages, snce a drect dervaton n terms of closed forms s not possble. Smlar to the SIR analyss, the ABEP for the SIR can be evaluated usng the CDF expressons for the fully correlated case provded n Table III. Based on these expressons and substtutng 39 for the.n.d. fadng as well as 4 for the..d. fadng case n 48 and employng numercal ntegraton the ABEP can be readly evaluated. Hgh SR Approxmaton: Substtutng 44 n 48, the followng closed-form approxmated expresson can be derved for the ABEP n the hgh-sr regme 1 + I P b α Γ + 1 D β. 5 }{{} D 4 From the last expresson we get G d =,G c = D 1/ 4. VII. UMERICA RESUTS AD DISCUSSIO In ths secton varous numercal performance evaluaton results, whch have been obtaned usng the prevous analyss, wll be presented. In partcular, results related to SISO as well as SIMO systems, SR- or SIR-based technques and dfferent K -dstrbuted..d. fadng and fully correlated shadowng condtons wll be presented and dscussed. For obtanng Fgs. 3 and 4, we have assumed = 5, = 1.6, = 5 db, whle we have used 36 and 4, respectvely. In both these fgures, the number of branches requred for achevng a predefned target OP s plotted, for dfferent values of the normalzed outage threshold, th / D. In Fg. 3, consderng SR-based SD, t s shown that the number of branches ncreases as th / D ncreases and/or the target OP decreases. Moreover, t s easly verfed that for relatvely low values of th / D and/or hgh target OP, t s not necessary to employ SD, snce even wth SISO the target OP s acheved. Ths means that the overall power consumpton of the recever sde

11 734 IEEE TRASACTIOS O COMMUICATIOS, VO. 63, O. 3, MARCH 15 Fg. 3. The number of branches as a functon of the OP and the normalzed outage threshold for SR-based SD recepton. Fg. 5. The ABEP for SISO and SIMO SD recevers wth SR-based and SIR-based technques, assumng DBPSK modulaton. Fg. 4. The number of branches as a functon of the OP and the normalzed outage threshold for SIR-based SD recepton. can be reduced by avodng unnecessary crcuty and channel estmatons. The same observatons hold also for the SIRbased SD scenaro, whch s depcted n Fg. 4. By comparng Fgs. 3 and 4, we observe that the SIR-based recever requres consderably less recepton branches than the SR-based one for obtanng the same target OP. Ths gan, however, comes at the cost of much hgher sgnal processng requrements for the SIR-based approach. Moreover, from these two fgures t can be also verfed that n most cases the predefned OP target s acheved by employng at most or 3 branches. Therefore, both SR-and SIR-based SD schemes represent excellent compromses as far as the performance and complexty or energy effcency tradeoff s concerned. Usng 47, 46 and 48, the ABEP of DBPSK s plotted n Fg. 5, as a functon of the average SR of the desred sgnal, D, for SISO, SR-based SD and SIR-based SD, respectvely. In ths fgure the parameters are taen as =, = 5 db and = 4. As expected, the best performance s provded by the SIR-based recever, whle the performance gap between SIR and SR-based SD ncreases as the number of dversty branches employed also ncreases. Addtonally, n the same fgure and for the hgher SR values,.e., > 5 db, the correspondng curves for the SR-and-SIR-based cases are plotted, usng 49 and 5, respectvely. In both cases the asymptotc expressons for the ABEP are qute close to the exact ones, especally for hgher values of average nput SR and/or small number of branches. Fnally, assumng..d. and not fully correlated shadowng condtons for the dfferent nterferng paths, the correspondng performance of a SISO system s also plotted. It s shown that the ABEP of the..d. case s always slghtly worst, verfyng thus that the fully correlated shadowng represents the worst case of shadowng. In Fg. 6, assumng = 5 db and = 4, the OP s plotted as a functon of the normalzed outage threshold under two communcaton scenaros, namely SISO system and SIR-based SD, for varous values of. We observe that the performance mproves as the number of branches ncreases, wth a decreased rate of mprovement though. A worth mentonng observaton that comes out of ths fgure s that as the nterferng sgnals shadowng parameter ncreases, the OP decreases. Ths s a reasonable result snce severe shadowng condtons n the nterferng sgnals result to a lower IR and thus to a hgher SIR. Addtonally, to better understand how nterference affects system s performance, an nterference lmted communcaton scenaro has been consdered by entrely neglectng nose effects. Therefore, n the same fgure, the correspondng OP performance for the SIR case has been depcted, usng the CDF expresson gven n Table III. In all cases, when

12 BITHAS AD ROTOGIAIS: MOBIE SYSTEMS I FADIG/SHADOWIG, OISE AD ITERFERECE 735 whlst n all cases the performance worsens wth the ncrease of. It s nterestng to note that for hgher values of D,the lne gap of the performances obtaned usng dfferent values of ncreases. For comparson purposes, computer smulaton performance results are also ncluded n Fgs. 5 7, verfyng n all cases the valdty of the proposed theoretcal approach. Fg. 6. The OP for SISO as well as SIR-based SD SIMO system. VIII. COCUSIO In ths paper, an analytcal framewor for evaluatng mportant statstcal metrcs of the nstantaneous output SIR of SISO as well as SIMO dversty recevers operatng over composte fadng channels has been presented. The proposed analyss s based on convenent expressons that have been extracted for the PDF and CDF of the sum of K -dstrbuted RVs, assumng dentcal, non-dentcal as well as fully-correlated dstrbuted parameters. Focusng on the latter case, varous statstcal characterstcs of SISO, SR-based SD and SIRbased SD recevers are derved n closed form, whch are then used to study system performance n terms of ABEP and OP. An asymptotc hgh SR analyss has been also presented based on whch the dversty and codng gan expressons are studed. The obtaned results ndcate that the combnaton of fadng/shadowng and nterference dsrupts serously the performance of the system. Furthermore, t s shown that a power effcent soluton that can consderably mprove ths stuaton s the employment of SD recepton wth a relatvely small number of dversty branches. APPEDIX A PROOF OF THEOREM 1 Δ The moments generatng functon MGF of = can be expressed as [11] M I s= exp Γ 1, A.1 s s s where Γα,x = x exp ttα 1 dt s the upper ncomplete Gamma functon [, eq. 8.35/]. The MGF of, for.n.d. nterference condtons, s gven by Fg. 7. The ABEP for SR-based SD recepton versus the number of nterferng sgnals, assumng DBPSK modulaton. M I s= M I s. A. nose s not present SIR case the performance shows an mprovement, as expected. An nterestng observaton though s that the performance gap between the SIR and SIR scenaros, manly depends on the number of dversty branches employed. Specfcally, the nose effects seem to play a more mportant role when SD recepton s used wth an ncreased number of antennas. In Fg. 7, consderng SR-based SD and assumng = 1 db, =, the ABEP s plotted as a functon of the number of nterferng sgnals, for varous values of the number of branches and dfferent average SRs of the desred sgnal D. We observe that as D and/or ncrease the ABEP decreases, Therefore, assumng non nteger values for, substtutng A.1 n A. and usng the nfnte seres representaton for the ncomplete gamma functon,.e., [, eq /], yelds M I s=g 1 1 s S [,1 S ] [ exp 1, s 1 ˆt h, where ˆt h, = t h,. Furthermore, snce h= 1 ˆt h, =1 + ˆt h,λn, n=1 λ 1,...,λ ] A.3 A.4

13 736 IEEE TRASACTIOS O COMMUICATIOS, VO. 63, O. 3, MARCH 15 z where = z x,y x=1 λ 1,...,λx be rewrtten as 1 y z x+1 z x+ λ 1 =1 λ =λ 1 +1 M I s= Γ1 1 S,1 s exp S, 1 + s, λ 1,...,λ z λ x =λ x 1 +1 n=1 ˆt h,λn, A.3 can. A.5 Captalzng on the power seres dentty = α x = β x = = c x, where c h = h = α β h gven n [, eq..316], A.3 can be smplfed as M I s= Γ1 1 S,1 s exp S, s 1 +, λ 1,...,λ h= G 3. A.6 The MGF expresson presented n A.6 s n a convenent form for applyng the nverse aplace transform gven n [44, eq ], leadng to 7, whch completes the proof. APPEDIX B PROOF OF THEOREM Consderng..d. nterference condtons the MGF functon of can be expressed as M I s=[ MI s ]. B.1 Assumng non nteger values for, substtutng A.1 n ths equaton, wth =, usng also the nfnte seres representaton for the ncomplete gamma functon,.e., [, eq /] and employng the bnomal dentty, B.1 can be wrtten as M I s= exp s s = 1 [ 1 Γ1 1 s h h= h!1 + h ] h. B. s Usng the useful power seres dentty provded n [, eq..314],.e., q= α qx q h = q= c qx q, wth c = α h, c m = 1 mα m q=1 qh m+qα qc m q for m 1, a smplfed expresson for B. can be derved as M I s=exp s = Γ1 1 [ ] G4 c h. B.3 h= s Based on the convenent expresson derved n B.3, the nverse aplace transform gven n [44, eq ] can be appled, and thus 9 s derved, whch completes the proof. REFERECES [1] P. S. Bthas and A. A. Rontoganns, Performance analyss of moble communcaton networs n the presence of composte fadng, nose and nterference, n IEEE Int. Symp. Sgnal Process. Inf. Technol., Athens, Greece, Dec. 13, pp [] V. Aalo and J. Zhang, Performance analyss of maxmal rato combnng n the presence of multple equal-power cochannel nterferers n a aagam fadng channel, IEEE Trans. Veh. Technol., vol. 5, no., pp , Mar. 1. [3] A. Annamala, C. Tellambura, and V. K. Bhargava, Smple and accurate methods for outage analyss n cellular moble rado systems-a unfed approach, IEEE Trans. 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Technol., vol. 54, no. 6, pp , ov. 5. [4] M. Xa and S. Assa, Impact of co-channel nterference on the performance of mult-hop relayng over aagam-m fadng channels, IEEE Wreless Commun. ett., vol. 3, no., pp , Apr. 14. [43] Y. Chen and C. Tellambura, Dstrbuton functon of selecton combner output n equally correlated Raylegh, Rcan, aagam-m fadng channels, IEEE Trans. Commun., vol. 5, no. 11, pp , ov. 4. [44] M. Abramowtz and I. A. Stegun, Handboo of Mathematcal Functons. ew Yor, Y, USA: Dover, 197. Petros S. Bthas S 4 M 9 receved the B.Sc. n electrcal engneerng and the Ph.D. degree wth specalzaton n wreless communcaton systems from the Unversty of Patras, Greece, n 3 and 9, respectvely. Durng 4 9, he was a Research Assstant at the Insttute for Space Applcatons and Remote Sensng of the atonal Observatory of Athens OA, Greece. Snce October 9 he s afflated wth the Department of Electroncs Engneerng of the Technologcal Educatonal Insttute of Praeus as a ab Instructor. Furthermore, durng the ovember of 1 and February 13 he was a Postdoctoral Researcher at the Department of Dgtal Systems, Unversty of Praeus. Snce the Aprl 13 he s afflated wth the Insttute for Astronomy, Astrophyscs, Space Applcatons and Remote Sensng of the OA. Hs has research nterests n the areas of cooperatve communcaton systems, cogntve relay networs and dstrbuted MIMO technques. Hs s author of 18 papers n nternatonal journals and more than n conference proceedngs. Dr. Bthas serves on the Edtoral Board of AEÜ Internatonal Journal of Electroncs and Communcatons Elsever. Dr. Bthas has been selected as an Exemplary Revewer of IEEE COMMUICATIOS ETTERS snce 1. He s also co-recpent of Best Paper Awards at the IEEE Internatonal Symposum on Sgnal Processng and Informaton Technology, 13. He s a member of the IEEE Communcatons Socety and the Techncal Chamber of Greece. Athanasos A. Rontoganns M 97 was born n efada Island, Greece, n He receved the 5 years Dploma degree n electrcal engneerng from the atonal Techncal Unversty of Athens TUA, Greece, n 1991, the M.A.Sc. degree n electrcal and computer engneerng from the Unversty of Vctora, Canada, n 1993, and the Ph.D. degree n communcatons and sgnal processng from the Unversty of Athens, Greece, n From 1998 to 3, he was wth the Unversty of Ioannna. In 3, he joned the Insttute for Astronomy, Astrophyscs, Space Applcatons and Remote Sensng IAASARS of the atonal Observatory of Athens OA, where he has been a Senor Researcher snce 11. Hs research nterests are n the general areas of statstcal sgnal processng and wreless communcatons wth emphass on adaptve estmaton, hyperspectral mage processng, Bayesan compressve sensng, channel estmaton/ equalzaton, and cooperatve communcatons. Currently, he serves at the Edtoral Boards of the EURASIP Journal on Advances n Sgnal Processng snce 8 and the EURASIP Sgnal Processng Journal snce 11. He s a member of the IEEE Sgnal Processng and Communcaton Socetes and the Techncal Chamber of Greece.