A GENERAL FRAMEWORK FOR CONTINUOUS TIME POWER CONTROL IN TIME VARYING LONG TERM FADING WIRELESS NETWORKS

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1 A GENERAL FRAEWORK FOR CONTINUOUS TIE POWER CONTROL IN TIE VARYING LONG TER FADING WIRELESS NETWORKS ohammed. Olama, Seddk. Djouad Charalambos D. Charalambous Elecrcal and Compuer Engneerng Deparmen Elecrcal and Compuer Engneerng Deparmen Unversy of Tennessee Unversy of Cyprus 158 ddle Drve 75, Kallpoleos Sree, P.O.Box 537 Knoxvlle, TN USA 1678 Ncosa, Cyprus ABSTRACT In hs paper, a general framework for connuous me power conrol algorhm under me varyng long erm fadng wreless channels s developed. Ths conrass mos of he power conrol algorhms nroduced n he leraure ha assume power conrol akes place n dscree me nervals and can only be used as long as he me duraon for successve adjusmens of ransmer powers s less han he coherence me of he channel. In connuous me power conrol, here s no resrcon on how fas he wreless channel s varyng. oreover, a suffcen condon for he exsence of he opmal connuous me power s derved. The opmal connuous me power conrol algorhm s developed under me varyng long erm fadng wreless channels, whch are based on sochasc dfferenal equaons. KEY WORDS odellng, power conrol, sochasc dfferenal equaon, Banach space, and long erm fadng 1. Inroducon Power conrol (PC s mporan o mprove performance of wreless communcaon sysems. The benefs of power mnmzaon are no jus ncreased baery lfe, bu also ncreased overall nework capacy. Users only need o expand suffcen power for accepable recepon as deermned by her qualy of servce (QoS specfcaons ha s usually characerzed by he sgnal o nerference rao (SIR [1]. The majory of research papers n hs feld use dscree me PC algorhms, whch can only be used as long as he me duraon for successve adjusmens of ransmer powers s less han he coherence me of he channel. In hs paper, a general framework for connuous me power conrol algorhm (PCA under me varyng (TV long erm fadng (LTF wreless channels s developed. In connuous me power conrol, here s no resrcon on how fas he wreless channel s me varyng. PCAs can be classfed as cenralzed and dsrbued. The cenralzed PCAs requre global ou-of-cell nformaon avalable a base saons. The dsrbued PCAs requre base saons o know only he n-cell nformaon, whch can be easly obaned by local measuremens. The power allocaon problem has been suded exensvely as an egenvalue problem for non-negave marces [1]-[], resulng n erave PCAs ha converge each user s power o he mnmum power [3]-[5], and as opmzaon-based approaches [6]. Sochasc PCAs ha use nosy nerference esmaes have been nroduced n [7], where convenonal mached fler recevers are used. I s shown n [7] ha he erave sochasc PCA, whch uses sochasc approxmaons, converges o he opmal power vecor under ceran assumpons on he sep-sze sequence. These resuls were laer exended o he cases when a nonlnear recever or a decson feedback recever s used [8]. uch of hs prevous work deals wh dscree me PCAs and sac me-nvaran channel models. In hs paper, a general framework for connuous me PCA under TV wreless channels s developed. In me-nvaran models, channel parameers are random bu do no depend on me, and reman consan hroughou he observaon and esmaon phase. Ths conrass wh TV models, where he channel dynamcs become TV sochasc processes [9]-[11]. These models ake no accoun relave moon beween ransmers and recevers and emporal varaons of he propagang envronmen such as movng scaerers. They exhb more realsc behavor of wreless neworks. In hs paper, we consder dynamcal TV LTF channel modellng. The dynamcs of LTF wreless channels are capured by sochasc dfferenal equaons (SDEs. The SDE model proposed allows vewng he wreless channel as a dynamcal sysem, whch shows how he channel evolves n me and space. In addon, allows welldeveloped ools of adapve and non-adapve esmaon and denfcaon echnques (o esmae he model parameers o be appled o hs class of problems [1]. The correc usage of any PCA and hereby he power opmzaon of he channel models, requre he use of TV

2 channel models ha capure boh emporal and spaal varaons of he wreless channel. Snce few emporal or even spao-emporal dynamcal models have so far been nvesgaed wh he applcaon of any PCA, he suggesed dynamcal model and PCAs wll hus provde a far more realsc and effcen opmal conrol for wreless channels.. Tme Varyng Lognormal Fadng Channel odel Wreless rado channels experence boh long-erm fadng (LTF and shor-erm fadng (STF. LTF s modelled by lognormal dsrbuons and STF are modelled by Raylegh or Rcean dsrbuons [13]. In general, LTF and STF are consdered as supermposed and may be reaed separaely [13]. In hs paper, we consder dynamcal modellng and power conrol for LTF channels whch are predomnae n suburban areas. The STF case has been consdered n [1]. The me-nvaran power loss (PL n db for a gven pah s gven by [13]: PL( d[ db] PL( d [ db] 1 log d = + α + Z (1 d where d d, PL( d s he average PL n db a a reference dsance d from he ransmer, α s he pah loss exponen whch depends on he propagaon medum, and Z s a zero-mean Gaussan dsrbued random varable, whch represens he varably of he PL due o numerous reflecons occurrng along he pah and possbly any oher uncerany of he propagaon envronmen from one observaon nsan o he nex. In TV LTF models, he PL becomes a random process denoed { X(, τ }, whch s a funcon of boh me, τ τ and locaon represened by τ, where τ = d/c, d s he pah lengh, c s he speed of lgh, τ = d /c and d s he reference dsance. The process { X(, τ }, τ τ represens how much power he sgnal looses a a parcular dsance as a funcon of me. The sgnal kx (, τ aenuaon s defned by S (, τ e, where k = ln ( 1 / [13]. The process X (, τ s generaed by a mean-reverng verson of a general lnear SDE gven by [9]: ( dx (, = β(, ( γ(, X (, d + δ(, dw(, ( σ X τ = N PL d db (, ( [ ]; ( where { W ( } s a sandard Brownan moon (zero drf, un varance whch s assumed o be ndependen X τ, N ( µ ; κ denoes a Gaussan random of (, varable wh mean µ and varance κ, and PL( d[ db ] s he average pah loss n db. The parameer γ (, τ models he average TV PL a dsance d from ransmer, whch corresponds o PL( d[ db ] a d ndexed by. Ths model racks and converges o hs value as me progresses. The nsananeous drf β (, ( γ(, X(, represens he effec of pullng β, τ represens he he process owards (, speed of adjusmen owards hs value. Fnally, δ (, τ γ τ, whle ( conrols he nsananeous varance or volaly of he process for he nsananeous drf. Defne { θ(, } { β(,, γ (,, δ (, }. If he random processes n { θ(, } are measurable and bounded, hen ( has a unque soluon for every X (, τ gven by [9]: β ([, ], X (, = e. X (, { β([ u, ], + e ( β( u, γ( u, du+ δ( u, dw( u where (3 β([, ], β( u, du. Ths model capures he emporal and spaal varaons of he propagaon envronmen as he random parameers { θ(, } can be used o model he me and space varyng characerscs of he channel. A every nsan of me X (, τ s Gaussan wh mean and varance gven by: [ τ ] [ τ ] β( [, ], E X(, = e. β( [ u, ], X + e β( u, γ ( u, du β( [ ],, Var X (, = e. β( [ u ] e ( u, du,, δ τ + σ oreover, he dsrbuon of S (, τ = e s lognormal wh mean and varance gven by: kx(, τ (4

3 [ τ ] + kvarx [ τ ] kex (, (, E[ S(, τ ] = exp Var S ke X k Var X ( ( ke [ X τ ] k Var [ X τ ] [ (, τ ] = exp [ (, ] + [ (, ] exp (, + (, The mean and varance n (4 and (5 show ha he sascs of he communcaon channel vary as a funcon of boh me and space τ. In hs paper, we consder he uplnk channel of a cellular nework and we assume ha users are already assgned o her base saons. Le be he number of mobles (users, and N be he number of base saons. The receved sgnal of he h moble a s assgned base saon a me s gven by: j j j j = 1 (5 y ( = p ( s ( S ( + n ( (6 where p ( s he ransmed power of moble j a me j, whch acs as a scalng on he nformaon sgnal s j (, ( s he channel dsurbance or nose a he base n saon of moble, and S ( s he sgnal aenuaon j coeffcen beween moble j and he base saon assgned o moble. Therefore, n a cellular nework he spaoemporal model descrbed n ( for mobles and N base saons can be descrbed as: ( (, = β (, γ (, (, + δj (, dwj (, ( ( σ dx X d j j j j ( X, = N PL d [ db] ;, 1, j j j and he sgnal aenuaon coeffcens Sj (, generaed usng he relaon Sj (, kxj (, τ e k = ln ( 1 / (7 τ are τ =, where. oreover, correlaon beween he channels n a mul-user/mul-anenna model can be nduced by leng he dfferen Brownan moons W j s T o be correlaed,.e., E W( W( = Q ( τ, where ( j W( W (, and Q ( τ s some (no necessarly dagonal marx ha s a funcon of τ and des ou as τ becomes large. The TV LTF channel models n (7 are used o generae he lnk gans of wreless neworks for he PCA proposed n he nex secon. 3. Sochasc Power Conrol Algorhm The am of he PCAs descrbed here s o mnmze he oal ransmed power of all users whle mananng accepable QoS for each user. The measure of QoS can be defned by he SIR for each lnk o be larger han a arge SIR. In hs secon, a PCA s nroduced based on he TV LTF channel models derved n he prevous secon. Consder he cellular nework descrbed n he prevous secon, he cenralzed PC problem for me-nvaran channels can be saed as follows [1]: mn p, subjec o ( p1,... p = 1 j pg pg ε, 1 + η j j where p s he power of moble, g j > s he menvaran channel gan beween moble j and he base saon assgned o moble, ε > s he arge SIR of moble, and η > s he nose power level a he base saon of moble. The generalzaon o (8 for he TV LTF channel models n (7, descrbed usng pah-wse QoS of each user over a me nerval [,T], s gven by [14]: mn, subjec o ( p1 (,... p ( p ( = 1 { ( ( (} ( ( ( + ( E p s S { k k k k } E p s S n ε ( where E{} s he expecaon operaor, [, T], and = 1,,. Defne, p F ( ( ( ( = E Sj ( ε ( E S ( ε E S ( η ( 11 ( ( η ( ( 1 1 p1 ε p u( E S,, p ( ε ( η ( E S (, = j, j (8 (9 (1

4 where, j = 1,,. Noe ha E Sj ( can be calculaed from (5 and (4. The consran n (9 can be rewren as: ( ( 1( ( p ( p ( F p p 1 (, ( ([, T]; 1 k < 1, p p C R (18 ( ( ( ( I F p u (11 If he power conrol problem n (9 s feasble, hen he opmal power sasfes: ( = ( ( + ( p* F p* u (1 where p *( s he opmal power vecor. Expresson (1 shows ha he opmal power s he fxed pon of he followng funcon: ( ( ( ( + ( Φ p F p u (13 The power vecor p ( s assumed o be connuous and bounded as funcons from [, T ] o + R, ha s, belongs o he Banach space of connuous and bounded funcons defned on [, T ], whch s denoed by C ([, T ] ; R, under he supremum norm gven by: b p ( = sup p ( (14 T = 1 Assumng u( C([, T] ; R, hen he map ( defned as: 1/ (: ([, ]; T ([, T]; Φ s Φ C R C R (15 The exsence of a fxed pon for Φ ( s guaraneed by he conracon mappng heorem or Banach s fxed-pon heorem [15], whch saes ha f ( ( ( ( k ( ( Φ p Φ p p p ( for some k < 1 and all 1(, ( ([, T]; p p C R, hen Φ ( has a unform fxed pon. Expresson (16 can be rewren as: ( ( ( ( k ( ( F p F p p p 1 1 (, ( ([, T]; p p C R 1 whch s equvalen o:, (17 Expresson (8 holds f and only f he followng holds: (( ( ( F p1 p sup k < 1 p1( p( p1( p( (, p ( C( [, T]; R p1 (19 The LHS n (19 s equal o he nduced norm of F ( vewed as a mulplcaon acng from C([, T ] ; R no C([, T ] ; R,.e., where ( ( k < 1 F ( ( ( F = sup F p p( C( [, T]; R (1 p( 1 F ( s equal o supremum wh respec o of he larges sngular value of F (, ha s ( σ ( ( F = sup F < 1 ( [, T] where ( F. Expresson ( gves a suffcen condon on he channels aenuaon coeffcens for he exsence of an opmal power. Expresson ( s sasfed f and only f: σ denoes he larges sngular value of ( ( ( 1, [, T] σ F < (3 Thus, f (3 s sasfed, he followng connuous me PCA wll converge o he mnmal power: p + = F p + u (4 k 1 ( ( k ( ( Noe ha n (4 ndex k s eraon on he connuous me power vecor and herefore does no represen he me varable as n mos PCAs n he leraure. Numercal resuls are presened n he nex secon. 4. Numercal Resuls The LTF cellular nework has he followng feaures: The number of ransmers (mobles s = 4.

5 The nformaon sgnal s ( = 1for = 1,...,. Targe SIR, ε ( = 5for = 1,...,. The ransmed powers are compued every 1 second, and he smulaon s performed for mnues. Inal dsances of all mobles wh respec o her own base saons are generaed as unformly ndependen dencally dsrbued (..d. random varables (r.v. s n [1 1] meers. Cross nal dsances of all mobles wh respec o oher base saons are generaed as unformly..d. r.v. s n [5-55] meers. The angle beween he drecon of moon of moble j and he dsance vecor passes hrough base saon and he moble j are generaed as unformly..d. r.v. s n [ 18] degrees. The average veloces of mobles are generaed as unformly..d. r.v. s n [4 1] km/hr. PL exponen s 3.5. Inal reference dsance from each of he ransmers s 1 m. PL a he nal reference dsance s 15 db. δ ( = 14 and β ( = 5 for he SDE s. η 's for = 1,..., are..d. Gaussan r.v. s wh zero mean and varance = 1-1 W. Fgure 1 shows he oal ransmed power of all mobles usng he proposed PCA n (4 for fxed me nsan ( = 4 seconds under he sochasc TV LTF wreless nework descrbed above. I can be noced ha he oal power converges o he opmal power afer few eraons even f he nal value s no close o he opmal one. Toal Power PCA Opmal Ieraon Fg. 1. Sum of ransmed power of all mobles for he proposed PCA under TV LTF wreless nework. Fgure shows he oal ransmed power of all mobles usng he proposed PCA n (4 as a funcon of me. I can be seen ha he oal power vares rapdly as a funcon of me, snce he mobles move n dfferen drecons and veloces. Toal Power Tme (mn Fg.. Sum of ransmed power of all mobles wh respec o me for he proposed PCA under TV LTF wreless nework. 5. Concluson A general framework for connuous me power conrol algorhm under me varyng long erm fadng wreless channels s developed. The channel models are represened by 1 s order sochasc dfferenal equaons (SDEs. The SDE models proposed allow vewng he wreless channel as a dynamcal sysem, whch shows how he channel evolves n me and space. In addon, allows well-developed ools of adapve and non-adapve esmaon and denfcaon echnques (o esmae he model parameers o be appled o hs class of problems. A suffcen condon on he me varyng channels aenuaon coeffcens for he exsence of he opmal connuous me power s derved. Snce he proposed power conrol algorhm consders connuous me power conrol, here s no resrcon on how fas he wreless channel s varyng. Numercal resuls show ha he proposed erave power conrol algorhm converges o he opmal power. Fuure work wll focus on dervng suffcen condon on he channel parameers { θ(, } for he exsence of he opmal connuous me power vecor. References [1] J. Zander, Performance of opmum ransmer power conrol n cellular rado sysems, IEEE Trans. on Vehcular Tech., 41(1, Feb [] J. Aen, Power balancng n sysems employng frequency reuse, COSAT Techncal Revew, 3, [3] N. Bambos and S. Kandukur, Power-conrolled mulple access schemes for nex-generaon wreless packe neworks, IEEE Transacons on Wreless Communcaons, 9(3, June.

6 [4] X. L and Z. Gajc, An mproved SIR-based power conrol for CDA sysems usng Seffensen eraons, Proc. of he Conference on Informaon Scence and Sysems, ar.. [5] G. J. Foschn and Z. ljanc, A smple dsrbued auonomous power conrol algorhm and s convergence, IEEE Trans. on Vehcular Tech., 4(4, Nov [6] S. Kandukur and S. Boyd, Opmal power conrol n nerference-lmed fadng wreless channels wh ouage-probably specfcaons, IEEE Transacons on Wreless Communcaons, 1(1,, [7] S. Ulukus and R. Yaes, Sochasc power conrol for cellular rado sysems, IEEE Trans. on Communcaons, 46(6, 1998, [8]. K. Varanas and D. Das, Fas Sochasc Power Conrol Algorhms for Nonlnear uluser Recevers, IEEE Trans. on Communcaons, 5(11, Nov., [9].. Olama, S.. Djouad, and C. D. Charalambous, Sochasc Power Conrol for Tme- Varyng Long-Term Fadng Wreless Neworks, EURASIP Journal on Appled Sgnal Processng, Arcle ID 89864, 13 pages, 6. [1] C. D. Charalambous, S.. Djouad, and S. Z. Denc, Sochasc power conrol for wreless neworks va SDE s: Probablsc QoS measures, IEEE Trans. on Informaon Theory, 51(, Dec. 5, [11] C. D. Charalambous and N. enemenls, General non-saonary models for shor-erm and long-erm fadng channels, Proc. of he EUROCO, Aprl, [1] L. Ljung and Sodesrom, Theory and pracce of recursve esmaon (IT Press, [13] T.S. Rappapor, Wreless communcaons: prncples and pracce (Prence Hall, [14] T. Hollday, A. J. Goldsmh, and P. Glynn, Dsrbued power conrol for me-varyng wreless neworks: Opmaly and convergence, Proc. of he Alleron Conference on Communcaons, Conrol, and Compung, Oc. 3. [15] B. Bolloba s, Lnear analyss (Cambrdge ahemacal exbooks, 199.