Selection Combining Upgrade Schemes for Mobile Units in Wireless Networks

Transcription

1 Selectio Combiig Upgade Scemes fo Mobile Uits i Wieless Netwoks GILL R. TSOURI ad DOV WULICH Depatmet of Electical ad Compute Egieeig Be-Guio Uivesity of te Negev Bee-Seva 845 ISRAEL tsoui@ee.bgu.ac.il Abstact: - Selectio combiig is cosideed fo upgadig te pefomace of mobile uits i wieless etwoks. Tee upgade scemes ae cosideed: ateas aay, ateas aay i cicula geometical settig ad ateas i liea geometical settig. Te poblem of divesity loss due to ateas poximity is addessed ad a appoac fo deceasig te loss is poposed. Te appoac is based o te Kaue-Loeve tasfom wic is deived fom te cael covaiace matix. As a example, te upgade of mobile uits i a IS-6 etwok is aalyzed fo Rayleig fadig caels. Pefomace is evaluated by computig te bit eo ate, outage pobability, aveage sigal to oise atio ad mea switcig ate of te cosideed scemes. Key-Wods: Divesity metods, fadig caels, wieless etwoks Itoductio Wieless etwoks opeate ude a costat demad fo ige tougput, due to te icease i uses ad multimedia sevices. Te developmet ad layout of ew etwok ifastuctue is costly ad upgade scemes ae pefeable if tey povide te adequate icease i capacity. A pomisig appoac to icease cael capacity ivolves te use of multiple ateas. Te optimal way of usig a multiple ateas eceptio aay is to combie te atea outputs usig Maximal Ratio Combiig (MRC) []. Tis appoac equies dow covesio of all ateas outputs to basebad ad good cael estimatio. MRC is ot suitable fo upgadig a existig eceive wic uses oly oe dow covesio cai fo a sigle atea. A alteative to MRC is Selectio Combiig (SC) [-7,]. SC scemes wee well eseaced i te past ad ecetly eceived eewed academic attetio [-4]. I SC te atea wit te igest Istataeous Sigal to Noise Ratio (ISNR) is cose by a Radio Fequecy (RF) switc ad is fed to a sigle dow covesio cai. It follows, tat SC scemes may offe a suitable solutio fo upgadig existig wieless etwoks by eplacig te sigle atea fot ed wit a multiple ateas fot ed ad te accompayig switc ad logic. I tis pape we aalyze tee SC upgade scemes. I may wieless eceives te ateas ae foced to be close to oe aote, suc is te case of mobile uits i a cellula etwoks. SC scemes suffe fom divesity loss we te ateas ae close togete. Liea aay RF fot ed KLT switc logic Double aay eceive Cicula aay Fig. Sceme of upgaded eceives Te loss is due to statistical coelatio betwee te ateas outputs [4-6]. Pefomace aalysis of dual bac SC scemes i coelated fadig caels is eadily available fo vaious types of caels. Howeve, fo lage aays ad abitay ateas settigs, pefomace aalysis is limited [-,7,]. Te followig pefomace paametes ae commoly cosideed: Bit Eo Rate (BER), outage pobability ad Aveage Sigal to Noise Ratio (ASNR). I tis pape we also examie te mea switcig ate. Tis paamete is impotat i te cotext of cellula etwoks wee te mobile uits

2 x z z z KLT (matix P) selecto logic Fig. Low pass equivalet model of te tiple ateas scemes ave limited powe supply. A ig switcig ate would icease powe cosumptio. Also, fo systems utilizig cotiuous pase modulatio, eac switc equies ealigmet of pase. Switcig ate cosideatio is give i [7] as well. A slow flat fadig cael is assumed. Fadig caels commoly cosideed ae Rayleig, Ricia ad Nakagami. I tis pape we coside Rayleig fadig. Tis fadig descibes a o lie of sigt ad ic scatteig sceaio ad is cosideed sevee. Fo te Rayleig fadig sceaio te covaiace matix may be detemied offlie ad teefoe te KLT is also kow offlie ad is costat fo a give caie fequecy ad atea settig. It sould be oted tat i te case of a asymmetic etwok, suc as a cellula etwok wee eac mobile uit is upgaded to multiple ateas ad te base statio emais wit oe atea, te duality atue of SC ca be used to select a atea fo tasmissio at te mobile uit. It follows, tat symmety i sigal quality is acieved. Tis appoac equies cael state ifomatio at te mobile uit. Suc ifomatio ca be deduced by te mobile uit we te same fequecy bad is used fo tasmissio ad eceptio ad may be tasmitted fom te base statio as side ifomatio otewise. I tis pape we popose a tecique fo educig te divesity loss icued by statistical coelatio of te ateas outputs. It ivolves de-coelatig te ateas outputs wit te Kaue Loeve Tasfom (KLT) [] pio to te switcig pocess. Te tecique equies te applicatio of a liea tasfomatio o te ateas outputs wic educes to a weigted sum of te aalog sigals. Te KLT depeds o te cael covaiace matix aloe ad is costat ad kow offlie fo Rayleig fadig caels. We ame SC afte KLT as Vitual Selectio Combiig (VSC). v eceive Te KLT is used extesively i applicatios wee a data set wit lage dimesios is to be educed fo pocessig. Suc is te case i ype spectal imagig teciques, wee a lage set of pictues is to be educed fo pocessig by a olie compute. Te KLT peseves te most of te ifomatio i te oigial set ad is teefoe used as a pepocessig tecique, commoly efeed to as Piciple Compoet Aalysis (PCA). Te motivatio fo usig KLT fo switced divesity metods suc as selectio combiig stems fom te obsevatio tat te selectio pocess is i fact a eductio i te dimesios of te obsevatio space ad it makes sese to apply PCA pio to te eductio. Tee SC scemes ae cosideed. Te fist uses a ateas aay. Te secod uses ateas i a cicula geometical settig (equally spaced o a cicle i space). Te tid uses ateas i a liea geometical settig (equally spaced o a lie i space). Te est of te pape is ogaized as follows. I sectio te upgaded scemes ae descibed matematically, i sectio te KLT is applied, i sectio 4 pefomace paametes ae evaluated, i sectio 5 esults ae aalyzed ad i sectio 6 coclusios ae daw. System Desciptio We coside tee SC scemes as depicted i fig.. Tee ateas ae laid out i a cicula o liea geometical settig. Te powe coelatio coefficiet (ρ) may be computed usig te followig fomula [8]: π ρ J d, () λ wee J is te zeo ode Bessel fuctio, d is te distace betwee te ateas ad λ is te caie wavelegt. Notice tat fo aow bad sigals ρ is pactically costat witi te bad. As a example, we assume tat te distace betwee cosecutive ateas is cm fo te double ateas aay,.5 cm fo te tiple ateas cicula aay ad 5 cm fo te tiple ateas liea aay. Tese sceaios fit a cellula wieless telepoy mobile uit wee tee is limited space fo atea placig. We coside a caie fequecy of 85MHz, as is te case i te IS-6 wieless stadad. It follows fom () tat te coelatio betwee eac pai of cosecutive ateas outputs is ρ. fo te double ateas aay, ρ.8 fo te tiple cicula aay ad ρ.67 fo te tiple liea aay.

3 . Tiple ateas scemes Te systems low pass equivalet fo te tiple ateas scemes is sow i Fig.. Notice tat placig te KLT ad switcig afte dow covesio fits te liea model. It is assumed tat te cael betwee te tasmit atea ad eac oe of te eceive ateas as flat (o-fequecy selective) fadig, i.e., its low pass equivalet may be epeseted by a vecto of complex valued cael atteuatios ( ) T,,, + j. Cosequetly, te eceived vecto of ateas outputs is E s x + z, wee x is a complex valued symbol to be tasmitted, E s is its eegy ad z is a vecto of additive, zeo mea complex oise assumed to be spatially i.i.d. wit equal vaiace N. We assume as te followig popeties: (A) },,, ; (A) E{ },,,; m,, ; m * E.5, i.e., E ( i ) ( i ) ( q ) m E m (A) { } { } { } E ; (A4) { } { } E. We use * to ote te cojugate opeatio ad E{ } to deote expectatio. Suc popeties ae caacteistic fo may wieless caels [8,9]. Note tat (A) meas tat te i-pase ad quadatue compoets ae otogoal ad tat (A) meas tat te ASNR is te same fo all atea outputs (omogeous baces). Te covaiace matix is ied as: Fo te cicula aay sceme: ρ H C E{ } ρ ρ ρ ρ ρ (a) Fo te liea aay sceme: ρ ρ C ρ ρ (b) ρ ρ We use H to ote te cojugate taspose opeatio. Te iitios i (a) ad (b) ae i accodace wit te costat coelatio model ad expoetial coelatio model accodigly as give i []. We assume positive values of C. Howeve, tey may be egative as well. I ay case, te give aalysis emais te same. Rayleig fadig meas tat as a Gaussia distibutio, wic togete wit (A)-(A4) meas tat te joit Pobability Desity Fuctio (PDF) of te eal valued adom vecto (,,,, ) T, as te followig fom: f ( x) wee (, x x ) ad C det T x, T x C x exp N ( π ) () x is te agumet of te PDF C is te covaiace matix of. Fom (A)-(A4) we obtai: C.5,,,, * wee E{ } follows tat, m m m m,,,,,,,,, (4), ad fom (A)-(A4) it also,.. Double ateas sceme Te deivatio fo te double ateas sceme is essetially te same as tat of te tiple ateas scemes ad is omitted fo bevity. We ie: ρ C (c) ρ Cael de-coelatio. Tiple ateas scemes Te vecto is compised of samples i space geeated by te aay outputs. Applyig te KLT, i te fom of te matix P, o esults i: v P P( ES x + z) (5) ~ E P x + P z S ~ E x + ~ z wee P ad ~ z P z. We use te followig otatio: ~ ~ ~ ~ ( ) T ~ ~ ) ( ),,, ad ( i ~ q + j. Accodig to te popeties of te KLT, te matix P H is a uitay oe ad its colums ae te eigevectos of C. It follows tat te poposed de-coelatio pocedue depeds oly o te cael covaiace matix. Te auto-covaiace matix of ~ is diagoal ad: C PC P H diag λ, λ, ~ ( λ S ), (6)

4 wee { } λ ae te eigevalues of C ad ae also te vaiaces of te vitual cael atteuatios. We may aage te eigevalues suc tat λ λ λ. P is also uitay, so te statistical popeties of ~ z ae te same as tose of z. Te sum of te eigevalues is equal to te sum of te vaiaces of te atteuatios i. Equivaletly, sice te oise vaiace is ucaged by P, te sum of te ASNRs of te compoets i is equal to te sum of te ASNRs of te compoets i v. I ote wods, te KLT distibutes te total ASNR uevely, so tat ucoelated o omogeous baces ae ceated. We teat te compoets i v as te outputs of vitual ateas. As te coelatio coefficiet ρ becomes lage (te distace betwee te ateas i te aay becomes smalle), te diffeece i ASNR betwee te vitual ateas iceases. It follows tat as ρ iceases te SC switc as bette baces to coose fom afte KLT ad ece a divesity gai is acieved wic couteacts te divesity loss. Sice te KLT is a liea tasfom ~ as a Gaussia distibutio ad te followig PDF is tue: T det C ~ x C ~ x ( ) exp f x, (7) N ( π ) wee te matix C ~ is te ivese covaiace matix ~ ~ ~ ~ ~ ~ ~,,,,,. All compoets of ~ ae mutually ucoelated ad λ λ λ. (8) C ~.5 λ λ λ of te vecto: ( ) T Fo te case we ρ.8 we obtai: Fo te cicula aay sceme: P (9a) λ.945, λ.945, λ.8 Fo te Liea aay sceme: P (9b) λ.65, λ.8, λ Double ateas sceme Te deivatio fo te double ateas sceme is essetially te same as tat of te tiple ateas scemes ad is omitted fo bevity. We obtai: P (9c) λ.6485, λ.55 Note tat P emais ucaged fo ay give ρ ad tat it ca implemeted by just summig ad subtactig te ateas outputs. 4 Pefomace evaluatio 4. Tiple ateas scemes We evaluate pefomace by computig BER, outage pobability (P out ), ASNR ad mea switcig ate (R SW ). Te outage pobability is ied as P out P ( SNR < t, SNR < t, SNR < t), () wee Es is te ISNR at te -t bac SNR N ad t is a miimal equied SNR eeded fo acceptable pefomace of te eceive. Note tat i view of (A.) SNR Es N. Te mea switcig ate is ied as [7] R π p () SW wee π is te pobability tat te selecto cooses atea ad p is te pobability to move fom atea to aote. Let us ie Π ( π ) T, π, π. BER depeds o te type of modulatio beig used. We aalyze te BER of BPSK as a example. Te classical appoac to obtai closed fom expessios of te pefomace paametes is fist to acieve a closed fom expessio of te ISNR at te switc output. Pefomace paametes ae te calculated i a staigtfowad fasio. A diffeet appoac is to use te momet geeatig fuctio [-4,]. I wat follows we obtai closed fom expessios fo most of te paametes ad use Mote Calo metods to obtai te otes. Pefomace of SC Te outage pobability may be witte as: SC P f ( x, x,., x5, x6 ) dxdx dx5dx () 6 out D D x, x : x + x <, wee D ( ) { },, ad N t is te ISNR tesold omalized by te Es ASNR of a sigle bac pio to KLT.

5 a c Fig. BER as fuctio of SNR i db a) Double ateas, ρ. b) Tiple cicula aay, ρ.8 c) Tiple liea aay, ρ.67 b To calculate R SW we use iitio (), i.e., we ave to fid p ad Π. We ave: p SC K ( x, x,., x5, x6 ) dxdx dxdx 5 6 dxi dxi f Di D wee D ( x, x ) () { : x + x },,,, {( x, x ): x + x < x + x }, i,, i Di i i i i, ad f ( ) is give by (). Sice te baces ae omogeous: SC SC p p,,., (4a) π i, i,, (4b) Usig te iitio fo R SW we ave: SC R SW. (5) Te ASNR is evaluated diectly as: SC ASNR f ( x, x,., x5, x6 ) xdx xdx x5dx5x6dx (6) 6 D D Te itegals () ad (6) wee umeically evaluated usig Mote Calo metods. Te esults wee veified usig compute simulatio of te low pass equivalet model. BER is evaluated usig compute simulatio of te low pass equivalet model. Statistics wee gateed fo 5 istaces ad bits pe cael. cael Pefomace of VSC Sice te baces ae ucoelated, aalytic calculatios fo VSC is easie ta fo SC. Usig (7), i te same mae as was doe fo SC, outie calculatios obtai: VSC ASNR exp j λ j λλλ λ + λ + λ + λ + λ + λ λλ λλ λλ λ + λ λ + λ λ + λ VSC P out (7) (8) R SW ad BER fo BPSK wee obtaied toug Mote Calo computatio ad validated wit compute simulatio as was doe fo BPSK BER of SC. expessios fo most of te paametes ad use Mote Calo metods to obtai te otes. 4. Double ateas sceme Recall tat pio to KLT te baces ae idetically distibuted ad statistically coelated ad tat afte KLT tey ae ucoelated but ubalaced i ASNR. We use te esults obtaied i [] fo P out

6 P out ASNR MRC t t exp exp t SC t exp L + + π ( ρ) π t( + ρ siϑ) exp ρ ( ) dϑ ( + ρ ρ siϑ) π + ρ a VSC exp o exp o t ( + ρ ) L t ( ρ ) ρ.5 + Tab. P out, ASNR fo te double ateas sceme b Fig.4 R SW at switc output as fuctio of ρ a) Double ateas, ρ. b) Tiple ateas-cicula aay, ρ.8 ; liea aay, ρ.67 ad ASNR of dual coelated ad ubalaced baces i Nakagami-m fadig wit te pope adjustmet to Rayleig fadig (m) ad system paametes. Tab.. summaized esults fo P out ad ASNR. R SW fo SC may be deived i te same mae as was doe fo te tiple ateas scemes. We obtai: SC R SW (9) R SW fo VSC may be deived by usig a Makov cai model as was doe i [7] fo fidig te SSC mea switcig ate fo ubalaced baces. Te deivatio is essetially te same ad is omitted fo bevity. We obtai: VSC ρ R SW () 5. Results I fig. BER as a fuctio of te ASNR at a sigle bac pio to KLT is sow. Te MRC cuve is displayed as a lowe boud, sice it is te optimal metod. Te sigle atea cuve epesets system pefomace pio to te suggested upgade. Te double atea ad tiple atea cuves epeset te pefomace of a sigle atea wit double ad tiple te eceptio suface accodigly. Tis is doe to evaluate te divesity gai acieved by usig multiple ateas istead of a sigle big atea ad elps evaluate te beefits of usig a aay. Fo te double atea sceme we fid tat te KLT offes o gai to te system ad tat a divesity gai is always acieved by SC compaed to te doubled suface atea. Notice, tat at db SC pefoms 5db bette ta te doubled suface atea ad tat SC pefoms oly ~db wose ta MRC. It follows tat fo te IS-6 stadad two ateas spaced cm apat ae a good upgade solutio fo mobile uits ad tat te divesity loss is sevee eoug to justify te use of KLT pio to switcig. Fo te tiple cicula aay sceme, we fid tat at low SNR SC offes o cosideable divesity gai as compaed to te tiple atea ad is actually outpefomed by te tiple atea by db. Howeve, VSC pefoms as te tiple atea. It follows, tat fo low SNR te KLT gais db i BER pefomace. Fo ig SNR SC gives cosideable divesity gais of 5db ad moe as compaed to te tiple atea. VSC always pefoms bette ta SC,

7 Fig.5 Outage pobability as fuctio of ρ fo te tiple cicula aay but pefoms almost te same as SC fo SNR>db. Te same teds ae appaet fo te tiple liea aay sceme, but sice ρ is sigificatly lowe te KLT offes egligible gais fo SNR>5db. It follows, tat te tiple atea SC scemes offe a good upgade solutio ad te KLT ca be used to potect te system fom divesity loss at low SNR. I fig.4 R SW is displayed as a fuctio of ρ. Fo all scemes, we fid tat altoug R SW is costat fo SC, R SW fo VSC deceases damatically as ρ iceases. Fo te tiple cicula sceme wit ρ.8 R SW is. compaed to.67 fo SC. Fo te tiple liea sceme wit ρ.67 R SW is. compaed to.67 fo SC. Fo te double ateas sceme wit ρ. R SW is.44 compaed to.5 fo SC. Fig.5-6 illustate te way i wic te KLT potects te system fom divesity loss. Te displayed esults ae fo te tiple cicula aay sceme wit vayig ρ (due to diffeet atea spacig ad/o diffeet caie fequecies). Te esults fo te tiple liea aay sceme ae vey simila to te esults fo te cicula aay sceme ad ae omitted fo bevity. Also, te esults fo te double bac sceme exibit te same teds o a smalle scale ad ae omitted as well to avoid epetitio. I fig.5 P out is displayed as a fuctio of ρ. P out is calculated wit te tesold beig te ASNR at a sigle bac pio to KLT. We fid tat as ρ iceases P out iceases fo SC. Howeve, fo VSC it is costaied low ad is almost te same as tat of a tiple atea fo all ρ. P out fo MRC ises to a poit wee it is almost te same as VSC. Tese esults illustate te way i wic te KLT costais divesity loss. I fig.6 te ASNR at te switc output is displayed as a fuctio of ρ. Notice tat MRC, beig te optimal appoac, pefoms as a tiple atea. Fig.6 ASNR at switc output as fuctio of ρ fo te tiple cicula aay We fid tat as ρ iceases SC exibits iceasig divesity loss to te poit wee it acts as a sigle atea. Howeve, fo VSC te ASNR ises as ρ iceases to te poit wee it is almost te same as MRC. 6. Coclusio Tee selectio combiig upgade scemes wee suggested fo eceives of mobile uits i wieless etwoks. Te upgade cosisted of eplacig te sigle atea fot ed wit a double o tiple ateas aay, followed by te Kaue Loeve tasfom ad a selectio combiig switc. Te Kaue Loeve tasfom depeds o te cael covaiace matix wic is kow offlie fo Rayleig fadig. Fo te upgade sceme based o tee ateas aaged i a cicula geometical settig wit a caie fequecy of 85MHz ad ateas spacig of.5 cm, as is expected fom a adeld mobile uit i te IS-6 stadad, te system suffes fom divesity loss ad te Kaue Loeve tasfom impoved BER pefomace fo low SNR by db. Te switcig ate is deceased damatically ad te aveage SNR is impoved as well. Fo te same system wit te ateas aaged i a liea geometical settig 5 cm apat a less damatic gai is acieved. Fo te upgade sceme based o a double ateas aay wit a caie fequecy of 85MHz ad ateas spacig of cm, te Kaue Loeve tasfom offes o cosideable gai. It was sow tat te Kaue Loeve tasfom potects selectio system fom divesity loss ad impoves outage pobabilities ad aveage SNR at selecto output.

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