Hidawi Publishig Corporatio EURASIP Joural o Wireless Commuicatios ad Networig Volume 9, Article ID 81478, 11 pages doi:1.1155/9/81478 Research Article Rate-Optimized Power Allocatio for DF-Relayed OFDM Trasmissio uder Sum ad Idividual Power Costraits Luc Vadedorpe, Jérôme Louveaux,Our Oguz, ad Abdellatif Zaidi Commuicatios ad Remote Sesig Laboratory, Uiversité Catholique de Louvai, Place du Levat, 1348 Louvai-la-Neuve, Belgium Correspodece should be addressed to Luc Vadedorpe, luc.vadedorpe@uclouvai.be Received 1 November 8; Revised 6 February 9; Accepted May 9 Recommeded by Eri G. Larsso We cosider a OFDM (orthogoal frequecy divisio multiplexig) poit-to-poit trasmissio scheme which is ehaced by meas of a relay. Symbols set by the source durig a first time slot may be (but are ot ecessarily) retrasmitted by the relay durig a secod time slot. The relay is assumed to be of the DF (decode-ad-forward) type. For each relayed carrier, the destiatio implemets maximum ratio combiig. Two protocols are cosidered. Assumig perfect CSI (chael state iformatio), the paper ivestigates the power allocatio problem so as to maximize the rate offered by the scheme for two types of power costraits. Both cases of sum power costrait ad idividual power costraits at the source ad at the relay are addressed. The theoretical aalysis is illustrated through umerical results for the two protocols ad both types of costraits. Copyright 9 Luc Vadedorpe et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial wor is properly cited. 1. Itroductio I applicatios where it is difficult to locate several ateas o the same equipmet, for size or cost issues, it has bee proposed to mimic multiatea cofiguratios by meas of cooperatio amog two or more termials. Cooperatio or relayig, also coied distributed MIMO, has gaied a lot of iterest recetly. Cooperative diversity has bee studied for istace i 1 3 (ad refereces therei) for cellular etwors. I this paper we cosider commuicatio betwee a source ad a destiatio, ad the source is possibly assisted with a relay ode. All the chaels (source to destiatio, source to relay ad relay to destiatio) are assumed to be frequecy selective ad i order to cope with that, OFDM modulatio with proper cyclic extesio is used. The relay operates i a DF mode. This mode is ow to be suboptimum 4, 5. Decode-ad-forward is adopted here as a relayig strategy for its simplicity ad its mathematical tractability. Two protocols (P1 ad P) are cosidered. Each protocol is made of two sigalig periods, amed time slots. The first time slot is idetical for both protocols. Durig this period, o each carrier, the source broadcasts a symbol. This symbol (affected by the proper chael gai) is received by the destiatio ad the relay. The relay may retrasmit the same carrier-specific symbol to the destiatio durig the secod time slot. Whether the relay does it or ot will be idicated by the optimizatio problem which is formulated ad solved i this paper. The protocol P differs from the protocol P1 i that, i the latter, the source does ot trasmit durig the secod time slot, irrespective to whether the relay is active or ot durig the secod time slot. For P, o a per carrier basis, the source seds a ew symbol if the relay is iactive. The reaso for ot havig the source ad the relay trasmittig at the same time is to avoid the iterferece that would occur i this case, thus rederig the optimizatio problem somewhat tedious. Moreover i practice source ad relay will have differet carrier frequecy offsets which is liely to require ivolved precorrectio mechaisms. A sceario with iterferece will be ivestigated i the future. For both protocols, wheever it is active, the relay uses the same carrier as the oe used by the source. This is a apriorichoicemadeheretomaetheoptimizatiomore tractable. It is however clear that carrier pairig betwee source ad relay is a topic for possible further optimizatio of the scheme. At the destiatio, it is assumed that for
EURASIP Joural o Wireless Commuicatios ad Networig the relayed carriers, the receiver performs maximum ratio combiig of what is received from the source i the first time slot, ad what is received from the relay i the secod oe, for each toe. OFDM with relayig has already bee ivestigated by some authors. I 6, the authors cosider a geeral sceario i which users commuicate by meas of OFDMA (orthogoal frequecy divisio multiple access). They propose a geeral framewor to decide about the relayig strategy, ad the allocatio of power ad badwidth for the differet users. The problem is solved by meas of powerful optimizatio tools, for idividual costraits o the power. I the curret paper, we restrict ourselves to a sigle user sceario but we ivestigate more deeply the aalytical solutio ad its uderstadig. We study power allocatio to maximize the rate for both cases of sum power ad idividual power costraits. We also compare two differet DF protocols ad show the advatage of havig the source also trasmittig durig the secod time slot. I 7 the authors cosider a setup which is similar to the oe we address i this paper but with oregeerative relays. I 8, the authors ivestigate OFDM trasmissio with DF relayig, ad a rate maximizig power allocatio for a global power costrait. They briefly ivestigate the power allocatio for the protocol amed P1 i the curret paper, ad a sum power costrait oly. O the other had they ivestigate optimized toe pairig. I 9, the authors cosider OFDM with multiple decode ad forward relays. They miimize the total trasmissio power by allocatig bits ad power to the idividual subchaels. A selective relayig strategy is chose. More recetly, i 1 the authors also cosider OFDM systems assisted by a sigle cooperative relay. The orthogoal halfduplex relay operates either i the selectio detectio-adforward (SDF) mode or i the amplify-ad-forward (AF) mode. The authors target the miimizatio of the trasmitpower for a desired throughput ad li performace. They ivestigate two distributed resource allocatio strategies, amely flexible power ratio (FLPR) ad fixed power ratio (FIPR). The paper is orgaized as follows. The system uder cosideratio is described i Sectio. The rate optimizatio for a sum power costrait is ivestigated i Sectio 3 for the two protocols. The cases of idividual power costraits are dealt with i Sectio 4. Fially umerical results are discussed i Sectio 5.. System Descriptio We cosider commuicatio betwee a source ad a destiatio, assisted with a relay ode. All lis are assumed to be frequecy selective ad this motivates the use of OFDM as a modulatio techique. Assumig that the cyclic prefix is properly desiged ad that trasmissio over all lis is sychroous, the scheme ca be equivaletly represeted by a set of parallel subsystems correspodig to the differet subchaels or frequecies used by the modulatio ad facig flat fadig over each li. The bloc diagram associated with the system for oe particular carrier (or toe) is depicted i Figure 1. P s () P r () λ sr () Source Relay λ sd () λ rd () Destiatio Figure 1: Structure of the system for carrier. Durig the first time slot, the source seds oe modulated symbol o each carrier. Durig the secod time slot, the relay selects some of the modulated symbols that it decodes, ad retrasmits them. For each relayed symbol, we costrai the relay to use the same carrier as that used by the source for the same symbol. Based o the two sigallig itervals, the destiatio implemets maximum ratio combiig for the carriers with relayig. As explaied earlier, we cosider two protocols, called P1 ad P. I protocol P1, the carriers that are ot relayed are simply ot used i the secod time slot (either by the relay or by the source). I protocol P, a ew carrier specific modulated symbol is set by the source i the secod time slot o each oe of the carriers that are ot used by the relay. Let us deote by A s () (resp.,a r ()) the amplitude of the symbol set by the source (resp., the relay) o carrier i the first (resp., secod) time slot, ad by λ sd () (resp., λ rd ()) the complex chael gai for toe betwee source (resp., relay) ad destiatio. The oise sample corruptig the trasmissio o toe durig the first time slot is s (), ad r () durig the secod period. These two oise samples are zero-mea circular Gaussia, white ad ucorrelated with the same variace. Deotig by s() the uit variace symbol trasmitted over toe, after proper maximum ratio combiig at the destiatio, the decisio variable obtaied at the th output of the FFT (Fast Fourier trasform) is give by r() = A s () λ sd () s() A r () λ rd () s() A s ()λ sd () s() A r ()λ rd () r(). The associated sigal to oise ratio is give by γ() = P s() λ sd () P r () λ rd (), () where we have used the followig otatios: P s () = A s () ad P r () = A r (). 3. Rate Optimizatio for a Sum Power Costrait We first ivestigate the case of a sum power costrait. The techiques used i this sectio will be useful i solvig the problem with idividual power costraits. It is well ow 11, 1 that the optimizatio with idividual power (1)
EURASIP Joural o Wireless Commuicatios ad Networig 3 costraits ca be solved by reformulatig it properly ito a equivalet problem with a sum power costrait. All chaels gais are assumed to be perfectly ow for the cetral device computig the power allocatio. The overhead associated with chael updatig is ot discussed further i the curret paper. We ivestigate the two protocols separately. 3.1. Protocol P1. For protocol P1, the rate achieved by the system for a duratio of OFDM symbols is give by 13: R = (1 P s() λ sd () ) S s S r mi { (1 P s() λ sr () ), (1 P s() λ sd () P r () λ rd () )}, (3) where S s is the set of carriers (or toes) receivig power at the source oly, ad S r the complemetary set, that is the set of carriers receivig power at both source ad relay. These sets are ot ow i advace ad must be characterized i a optimal way. I 13 the sigal to oise ratio without fadig was assumed to be symmetric throughout the etwor. Here the model is more geeral ad otatios are itroduced to possibly allow differet trasmit powers at the source ad at the relay, ot oly for the same carrier but also for differet carriers. For a relayed carrier, assumig a decodead-forward mode, the rate is the miimum betwee the rate o li s d ad the rate o the li s r. The power allocatio which maximizes (3)isfirstivestigatedforasum power costrait N t =1 P s () P r () P t, (4) where P t is the total power budget available for the source ad the relay together, ad N t is the total umber of carriers. Below, the objective fuctio will be wored out i order to fid criteria eablig to decide about the set S s or S r to which each carrier has to be assiged. The Lagragia for the optimizatio of the rate, taig ito accout the total power costrait ad the decode-adforward costraits, is defied by L 1 = i (1 P s() λ sd () ) (1 i ) (1 P s() λ sd () P r () λ rd () ) μ i P s () (1 i ) P s () P r () P t ρ (1 i ) P s () λ sd () P r () λ rd () where μ is the Lagrage multiplier associated with the global power costrait ad ρ is the Lagrage multiplier associated with the decodability (perfect decode ad forward) costrait o carrier. Thei are idicators taig values or 1 ad whose optimizatio will provide the solutio for the assigmet to sets S s (i = 1) ad S r (i = ). Let us first ivestigate whether the decodability costraits are active or ot for relayed carriers. For relayed carrier q, i q =. If a costrait is iactive, its associated Lagrage multiplier is zero 14. Assumig this may be the case, settig the ρ q = ad taig the derivative of the Lagragia with respect to the powers for a relayed carrier leads to P s ( q ) = P r ( q ) = (1 P s(q) λ sd (q) P r (q) λ rd (q) λ sd (q) = μ, (1 P s(q) λ sd (q) P r (q) λ rd (q) λ rd (q) = μ. ) 1 ) 1 This shows that assumig that the costrait is ot saturated, the equatios lead to λ sd (q) = λ rd (q).thisimposes a costrait o the curret chael state, which is almost certai ot to happe. Hece, except i very margial cases, the decode-ad-forward costrait has to be saturated. This meas P s () λ sr () = P s () λ sd () P r () λ rd (), P s () = λ rd() P r () λ sr () λ sd () = α()p r(), where the last lie defies the coefficiet α(). Hece for relayed carrier, the total amout of power P() allocated to that carrier will be give by P() = P s () P r () = (1 α()) P r () = P s ()(1 α())/α(). Therefore the Lagragia ca be writte as: L = i (1 P() λ sd() ) (1 i ) (1 P() λ sr() μ i P() (1 i ) P() P t, ) α() 1α() (6) (7) (8) P s () λ sr (), (5) where for S s, P() = P s () adp r () =, while for S r, P() = P s ()P r () withp s () = α() P r ().
4 EURASIP Joural o Wireless Commuicatios ad Networig The solutio for the carrier assigmet ca be foud by taig the derivatives with respect to the idicators. We have that 1 ( P ( q ) λ sd (q) ) / = i q 1 ( P ( q ) λ sr (q) ) (α / ( ) ( ( ))), q / 1α q >, i q = 1, <, i q =. (9) It appears that whe λ sd (q) α( q ) 1α ( q ) λ sr (q) (1) the carrier should have i q = adbeallocatedtosets r.by oppositio, whe λ sd (q) α( q ) 1α ( q ) λ sr (q) (11) the carrier should be allocated to set S s. Ivestigatig (3) it should be clear that whe oe has λ sr (q) λ sd (q), because of the mi, the rate obtaied by allocatig the carrier to the set S s will always be higher tha the rate obaied if the carrier were allocated to S r.it is worth otig that, if λ sr (q) λ sd (q), the iequality betwee (α(q)/(1α(q))) λ sr (q) ad λ sd (q) is equivalet to the iequality betwee λ rd (q) ad λ sd (q).asamatter of fact, with the defiitio of α(q), α ( q ) 1α ( q ) λ sr (q) = λ sr (q) λ rd (q) The, λ sr (q) λ rd (q) λ sr (q) λ sd (q) λ rd (q) λ sd (q), λ sr (q) λ sd (q) λ rd (q). λ sr (q) λ rd (q) λ sr (q) λ sd (q) ( λ rd (q) λ sd (q) ) λ sd (q), λ sr (q) ( ( ) λ rd q ( ) λ sd q ) The above shows that λ sd ( q ) ( λ rd ( q ) λ sd ( q ) ). (1) (13) λ sd (q) α( q ) 1α ( q ) λ sr (q) λ sd (q) λ rd (q). (14) This meas that whe λ sr (q) λ sd (q), the allocatio to S s or to S r of the carrier may be based o either comparisos i (14) because they are equivalet. Ad i short, to be relayed, a carrier should fulfil the followig two coditios simultaeously: λ sr (q) λ sd (q) ad λ rd (q) λ sd (q). Now that the assigmet is ow, the Karush-Kuh- Tucer (KKT) optimality coditios are such that, at the optimum, for S s, P() = P() 1 λ sd () = μ (15) for the carriers to be served, ad for carrier such that <μ P() (16) the power should be set to P() =. For carriers S r ad to be served with power, P ( q ) = P() 1 1α() λ sr () = μ α() (17) while if P ( ) <μ q (18) we should set P(q) =. All these derivatios basically also show that, after the assigmet step, our costraied optimizatio problem ca actually be solved thas to the semial waterfillig algorithm, applied to a water cotaier built either from / λ sd () or from (/ λ sr () )((1 α())/α()). The latter values actually show that the costrait related to the DF operatig mode of the relay leads to particular values to be used for the cotaier. More specifically, for the set S r, these values are modified values with respect to the λ sr (). 3.. Protocol P. I this case, the rate achieved by the system over a duratio of OFDM symbols is give by 13: R = ( S s S r mi 1 P s() λ sd () ) { (1 P s() λ sr () ), (1 P s() λ sd () P r () λ rd () )}, (19) where S s is the set of carriers (or toes) receivig power at the source oly, ad S r is the complemetary set, that is, carriers receivig power at both the source ad the relay. We also deote by P s () the power allocated to a carrier at the source. If this carrier is ot relayed, each protocol istat uses P s ()/. Aalysis of this objective fuctio shows that the DF costrait is also saturated o all carriers usig the relay, lie
EURASIP Joural o Wireless Commuicatios ad Networig 5 for protocol P1. Hece for a relayed carrier with a allocated power P(q) the rate evolves as ( ( ) R r q = 1 P( q ) λ sr () α ( q ) ) 1α ( q ). () For a orelayed carrier q, ad a total allocated power P(q) (over the two istats), the rate evolves as ( ) R s q = (1 P(q) λ sd (q) ) (1) Whe λ sd (q) > λ sr (q) (α(q)/(1 α(q))) we have that R s (q) >R r (q) for ay value of P(q). O the cotrary, whe λ sd (q) < λ sr (q) (α(q)/(1 α(q))) we have R s (q) <R r (q). HoweverthisisolyvalidforP(q) λ t where λ t = 4 λ sr (q) ( ( ) ( ( ))) α q / 1α q λ sd (q) λ sd (q) 4. () If P(q) λ t, eve whe λ sd (q) < λ sr (q) (α(q)/(1α(q))), the power is better used by allocatig the carrier to set S s. Let us defie the followig Lagragia, with a Lagrage multiplier μ associated with the global power costrait, ad taig ito accout the saturatio of the DF costraits: L 3 = R μ P s () P() P t (3) S s with R = ( S s S r 1 P s() S r λ sd () ) (1 P() λ sr() ) α(). 1α() (4) Equatig to the derivatives of this Lagragia with respect to the power, we get for S s, 1 P s () = μ λ sd (), (5) where stads for max,.. Similarly, for S r, 1 P() = μ 1α() λ sr (). (6) α() Agai the derivatios show that the costraied optimizatio problem ca be solved usig the waterfillig algorithm, applied to a water cotaier built either from / λ sd () or from (/ λ sr () )((1 α())/α()). It is also importat to ote that for the orelayed carriers two idetical values have to be used for the water cotaier, correspodig to the two protocol istats. At the ed of the waterfillig oe checs if ay of the relayed carriers receives a amout of power larger tha the threshold give by (). If this happes, the relayed carrier fulfillig this coditio ad for which the rate icrease is the largest oe is moved from the set S r to the set S s. The waterfillig is applied agai. This procedure is iterated till oe of the relayed carrier receives a amout of power larger tha its associated threshold. I the sequel this procedure will be amed the reallocatio step. 4. Rate Maximizatio for Idividual Power Costraits This sectio is devoted to the power allocatio which maximizes the rates uder idividual power costraits o the source ad the relay respectively: N t =1 N t =1 P s () P s, (7) P r () P r. (8) First, ote that for the optimum power allocatio with idividual power costraits, it might happe that costrait (8) is iactive for certai values of chael parameters, but costrait (4) will always be active. I other words, at the optimum, the full available power will always be used at the source, while some of the power available at the relay may ot be used. This ca be explaied usig simple ituitive argumets. Assume a solutio is foud such that P s is ot fully used. The rate ca be further icreased by allocatig the remaiig source power to a carrier i set S s or i set S r. For the relay power, thigs may be differet. For istace, it may eve happe that all carriers are allocated to the set S s i which case the relay does ot trasmit at all. Oe way to tae this particular case ito accout is to perform a first optimizatio (called first step hereafter), tryig to allocate the source power i a optimum way, ot cosiderig the costrait o the relay power. After this allocatio process of the source power, oe has to chec whether the relay power is sufficiet or ot. If it is sufficiet, the the optimum solutio correspods to this particular situatio i which the full relay power is ot used. If ot, it ca ow be assumed that the relay power costrait is satisfied with equality at the optimum, ad the full iterative method explaied below should be used. Let us first describe the first step. 4.1. First Step. Agai, we aalyze the two protocols separately. 4.1.1. Protocol P1. The problem i this case is still to maximize (3) where it is ow assumed that the costrait o P r may ot be active. This meas that there is eough relay power such that for a relayed carrier, P r () ca always be made large eough to have P s () λ sd () P r () λ rd () P s () λ sr (). (9) As discussed above, the costrait o the source power beig saturated the associated Lagrage multiplier μ s may be differet from. Here we ivestigate a solutio for the case where the relay power is ot saturated ad the related
6 EURASIP Joural o Wireless Commuicatios ad Networig Lagrage multiplier is the. The correspodig Lagrage fuctio ca be writte as: L 4 = (1 P s() λ sd () ) S s S r (1 P s() λ sr () ) μ s P s () P s S s (3) I agreemet with the idicator variables used above, whe λ sd (q) λ sr (q) carrier q should be allocated to set S s. I the reverse case, it should be allocated to set S r. Oce the assigmet is ow, taig the derivative with respect to P s (q)withq S s ad equatig it to, it comes ( ) = D q (P s ) = P s (q) 1 P s q λ sd (q) = μ s. (31) For a carrier q i the other set, S r,weget ( ) = D P s q q(p s ) = P s (q) 1 λ sr (q) = μ s. (3) Hece the problem ca be solved by meas of a waterfillig procedure, where the cotaier is built from values / λ sd (q) i set S s,advalues/ λ sr (q) i set S r.with such a allocatio procedure, the miimum power required at the relay is give by S r P r (q) wherep r (q) = P s (q)/α(q). If this value is below the power available at the relay, the problem is solved. This would correspod to a situatio where the relay is located far away from the source, ad, i a sese, ot very useful for the protocol used here. Otherwise oe has to ivestigate the situatio where both power costraits are active (saturated), which is of most iterest. 4.1.. Protocol P. The correspodig Lagrage fuctio ca be writte: L 5 = ( 1 P s() λ sd () ) S s S r (1 P s() λ sr () ) μ s P s () P s S s (33) Taig the derivative with respect to P s (q) withq S s ad equatig it to, it comes Ps (q) ( ) = D q (P s ) = 1 P s q λ sd (q) = μ s. (34) For a carrier q i the other set, S r, ( ) = D P s q q(p s ) = P s (q) 1 λ sr (q) = μ s. (35) So the coclusios are similar to those draw for protocol P1. The problem ca agai be solved by meas of a waterfillig procedure, where the cotaier is built from the values / λ sd (q), ad the values / λ sr (q) i set S r.howeverit has to be oted that for the values related to set S s those values have to be used twice because of the two time slots. Besides that, the reallocatio procedure has to be implemeted: it has to be checed whether ay of the carrier allocated to set S r receives a amout of power above a certai threshold. If this happes, carriers have to be moved from set S r to set S s,ad the waterfillig has to be applied till this o loger happes, as explaied above. The value to be used for the threshold is similar to (), where λ sr (q) has to be used istead of λ sr (q) α(q)/(1 α(q)). 4.. Secod Step. A secod step is eeded uless the power used at the relay by the procedure described i the first step is below the available relay power. Two Lagrage multipliers, μ s ad μ r, ow have to be used for the power cotraits. Oe elemet i the directio of the solutio lies i the observatio 1 that the rate oly depeds o the products of powers ad (possibly modified) chael gais. Hece allocatig power P to a carrier with gai λ provides the same rate as allocatig power μp to a carrier with gai λ /μ. Let us assume for the momet that the optimum μ s ad μ r are ow. The allocatio rules proposed above to defie the sets S s ad S r should be revisited with gais modified as: λ μ sd = λ sd /μ s ; λ μ sr = λ sr /μ s ad λ μ rd = λ rd /μ r. The equivalet powers uder cosideratio are ow P μ s (q) = μ s P s (q)adp μ r (q) = μ r P r (q). 4..1. Protocol P1. Let us defie the followig Lagragia: L μ 1 = i 1 Pμ s () sd () (1 i ) 1 Pμ s () sd () μ Pr () rd () Ps μ () μ s P s (1 i ) Pr μ () μ r P r ρ (1 i ) P μ s () sd () μ Pr () rd () S r Ps μ () λ μ sr(). (36) It is iterestig to compare this Lagragia with the oe give by (5). Actually they both have the same structure. The
EURASIP Joural o Wireless Commuicatios ad Networig 7 first differece is that (5) isbasedop s ad λ s while (36) is based o P μ s ad λ μ s. Assumig that μ s ad μ r are ow, ad thas to the use of the modified gais ad powers, the idividual power costraits give rise to a sigle sum power costrait. The associated Lagrage multiplier ow has to be equal to 1. Based o these observatios, it turs out that for fixed μ s ad μ r all the results derived i Sectio 3 apply to our problem with idividual power costraits, ad to the powers ad the gais that have bee properly ormalized. I particular it ca be cocluded that for the carriers usig the relay, the decode-ad-forward costrait will be saturated, leadig to P μ r (q) = P μ s (q)/α μ (q). Hece P μ r (q) adp μ s (q) should be allocated simultaeously leadig to a total power deoted by P μ (q) = P μ r (q) P μ s (q) = (1 α μ (q))p μ r (q) = P μ s (q)(1 α μ (q))/α μ (q)where α μ( q ) rd (q) = λ μ sr(q) sd (q) = μ s α ( q ). (37) μ r Cosiderig that P μ (q) = Ps μ (q)(1 α μ (q))/α μ (q), we also have Ps μ ( ) μ q λ sr(q) = P μ( q ) μ λ sr(q) α μ( q ) 1α μ( q ) = P μ () λ μ sr() μ s α() μ r α()μ s = P μ( q ) μ λ sr(q) α ( q ) μ r μ s α ( q ). (38) Therefore, omittig the idicators, the Lagragia ca be rewritte as L μ = 1 Pμ s () sd () S s S r sr() 1P μ () P μ s () μ s α() μ r α()μ s P μ () μ s P s μ r P r (39) Carrier q should be placed i set S s if sd (q) λ μ sr(q) α μ( q ) 1α μ( q ) = λ sr (q) ( ) α q ( μr α ( q ) ). μ s (4) Based o the above, ad relatios (14) tobeadaptedwithλ μ ad α μ it turs out that the selectio rule whe λ μ sd (q) λ μ sr(q) amouts to choosig S s whe λ μ sd (q) λ μ rd (q) or whe λ sd (q) λ rd (q) μ s (41) μ r ad vice-versa. Therefore, the allocatio procedure of the carriers turs out to be equivalet to that i the sum power case, with properly modified chael gais. There is however oe importat exceptio to this rule which is related to the particular case where the equality λ sd (q) = λ rd (q) holds. It has bee assumed previously that this particular case eeds ot beig ivestigated as it is very uliely to happe. This applies for the sum power costrait. However, i the case of idividual power costraits, the procedure is ow worig with the modified values λ μ (q) which are o loger give but deped o the Lagrage parameters μ s ad μ r. It may happe (ad has bee ecoutered for some of the chaels radomly geerated) that the optimal values of these Lagrage parameters are such that the equality is exactly met o some carriers (usually at most oe). This particular situatio eeds a few additioal developmets ad adjustmets which have bee preseted i 15 ad will ot be repeated here. For a carrier belogig to the set S s, the rate gai ad optimality coditios are give by 1 Ps μ ( ) = Ps μ (q) q sd (q) = 1. (4) This leads to Ps μ ( ) q = 1 μ s λ sd (q). (43) For a carrier belogig to the set S r, the gai ad optimality coditios are give by P μ( q ) = P μ (q) 1 λ sr (q) μ s α(q)μ r = 1. α(q) (44) The correspodig power allocatio is give by P μ( q ) = 1 λ sr (q) μ r α(q)μ s. (45) α(q) So far, we have assumed that μ r ad μ s were ow. I fact there is a sigle pair (μ s, μ r ) for which the two power costraits are simultaeously fulfilled. To fid this pair, the followig algorithm is proposed. The idea is to sca all possible assigmets to sets S s ad S r. For carriers such that λ sd (q) λ sr (q), as discussed above, the carrier will be assiged to set S s. For the other carriers, with λ sd (q) λ sr (q), relayig may be cosidered. Equatio (41) says that the assigmet of a carrier cadidate for relayig depeds o the ratio λ rd (q) / λ sd (q). By sortig the carriers cadidates for relayig by decreasig order of the ratios λ rd (q) / λ sd (q), all possible assigmets ca be cosidered. As a matter of fact, if a sigle carrier gets relayed it will be the first oe i the sorted set. If two get relayed, it will be the first two, ad so forth. Therefore, by cosiderig all possible sets of first carriers i this sorted set, all possible assigmets ca be ivestigated. We have as may
8 EURASIP Joural o Wireless Commuicatios ad Networig situatios to cosider as we have carriers beig cadidates to be relayed. For each situatio, the assigmet to sets S s ad S r is fixed. For a fixed assigmet, the optimizatio problem to be solved is covex. The correspodig dual problem is also covex. The dual problem ca be solved by taig the derivativesofthedualobjectivewithrespecttoμ s ad μ r, ad equatig these derivatives to zero. The values of μ s ad μ r solvig these equatios ca be etered i the primal problem, ad the optimum power values ca be obtaied. The problem is that the equatios to fid the optimum μ s ad μ r are oliear. They ca be solved for istace i a iterative maer. These derivatives with respect to μ s ad μ r correspod to the two power costraits that have to be fulfilled. Hece ay classical method ow to fid the roots of a fuctio (herethederivativeswithrespecttoμ s ad μ r )cabeused. A typical method used is the so-called subgradiet method where the correctio to the Lagrage variables μ s ad μ r at step i is made proportioally to the error o the costraits. Here we try to improve this classical method by usig a Newto-Raphso algorithm where the first derivative of the objective fuctio (here the objectives are the costraits) is also used. A Newto-Raphso approach is ow to have quadratic covergece, ad to always coverge for a covex objective fuctio. At iteratio i, the power prices μ r ad μ s are updated accordig to 1 q P s (q) q P s (q) μ s q P r (q) μ s ( ) P s q Ps q ( ) P r q Pr μi1 s = μi s λ μ i1 r μ i r q μ r q P r (q) μ r (46) This Newto-Raphso procedure is thus to be repeated for each oe of the possible assigmets. 4... Protocol P. Adaptig the results of the previous subsectio leads to the followig Lagragia with the modified gais ad powers: L μ 3 = 1 Pμ s () sd () S s S r sr() 1P μ () P μ s () μ s α() μ r α()μ s P μ () μ s P s μ r P r (47) For a carrier belogig to the set S s, the rate gai ad optimality coditios are give by 1 Ps μ ( ) = Pμ s (q) q sd (q) = 1 (48) which leads to Ps μ ( ) q = 1 μ s λ sd (q). (49) For a carrier belogig to the set S r, the gai ad optimality coditios are give by P μ( q ) = P μ (q) 1 λ sr (q) μ s α(q)μ r = 1. α(q) (5) The correspodig power allocatio is give by P μ( q ) = 1 λ sr (q) μ r α(q)μ s. (51) α(q) Equatios (49) ad(51) also show that the powers are give by a waterfillig procedure with a commo water level 1 or a commo power costrait, ad cotaiers defied by these equatios. The problem is agai equivalet to the sum power case ad the procedure defied for the maximisatio problem i Sectio 3. ca be reused. The λ sd (q) have to be replaced by λ sd (q) /μ s, ad the λ sr (q) α(q)/(1 α(q)) by λ sr (q) α(q)/(μ r α(q)μ s ). The commets about the allocatio of the carrier to set S s or S r are the same as i the case of protocol P1. Recall also that the reallocatio step has to be implemeted. The Newto-Raphso procedure for the updatig of μ s ad μ r is similar to that used for protocol P1. 5. Results I order to illustrate the theoretical aalysis, umerical results are provided ad discussed. The umber of carriers is set to N t = 18. Chael impulse resposes (CIR) of legth 3 are geerated. The taps are radomly geerated from idepedet zero mea uit variace circular complex gaussia distributios. Hece the power delay profile is flat. All taps have a uit variace for all lis. From these CIRs, FFT are computed to provide the correspodig λ xy (x {s, r}, y {r, d}). We set = 1. For illustrative purposes, results are first preseted for oe particular chael realizatio. The power is set to P t = for the sum power costrait, ad to P s = 1 ad P r = 1 for the case of idividual power costraits. Figure shows the gais λ sr () (solid curve), λ sd () (dash-dotted), λ rd () (dashed) i dbw of the chaels. Figure 3 shows, for protocol P1 ad the sum power costrait, the result about the power allocatio ( ) ad the possible additioal split wheever relevat amog source power (solid lie) ad relay power (dashed). The s idicate whether the relay is active ( at the top of the figure) or ot ( i ). I this case, the power used by the source is 136 ad that by the relay is 64. The total rate obtaied here is 75.45 bits per a duratio of OFDM symbols. If preferred, this rate N b (bits) per OFDM symbols may readily be coverted to a spectral efficiecy by computig N b /N t (1β) (bits/sec/hz) where β is the roll-off factor. Figure 4 reports the power allocatio for protocol P with a sum power
EURASIP Joural o Wireless Commuicatios ad Networig 9 15 Frequecy resposes of the differet chaels (db) 3.5 Power allocated to source ad to relay 1 3 5.5 (db) 5 1.5 1 1 15.5 4 6 8 1 1 14 Carrier positio LSR LSD LRd Figure : Gais λ sr (), λ sd (), λ rd () i dbw. 4 6 8 1 1 14 Carrier positio Total power Source power Relay power Relay idic Figure 4: Fial power allocatio to source ad relay i the sum power case for protocol P..5 Power allocated to source ad to relay 16 P1P: rate versus power sum optimum /uiform LSR 1 LSD LRD 1 14 1 1.5 1 Rate (bits) 1 8 6.5 4 4 6 8 1 1 14 Carrier positio Total power Source power Relay power Relay idic Figure 3: Fial power allocatio to source ad relay i the sum power case ad protocol P1. 5 1 15 5 3 P t (dbw) P1 opt P opt P1 uif P uif Figure 5: Rate versus P t (dbw) for the two protocols ad uiform ad optimized power allocatio for the sum power costrait. Taps of the λ sr () have a variace dbs above those associated with the λ sd () ad the λ rd (). costrait. Recall that for a orelayed carrier the amout of source power show has to be used twice: oce per time slot. The rate achieved for the particular chael realizatio uder cosideratio here is 377.45 bits for a duratio of OFDM symbols. It is also iterestig to metio that i this case, the power allocated to the source for the chael realizatio uder cosideratio is 186.8 ad to the relay, the remaider meaig 13.. Compared to protocol P1, the gai is oticeable ad is clearly due to the better exploitatio of the secod time slot. With protocol P1 ad idividual power costraits, the bit rate achieved is 39.74 bits for a duratio of OFDM symbols. Compared to the same protocol with the sum power costrait, the observed rate loss is due to the values chose here for the idividual power costraits (1-1) which are rather differet from the values devoted to the two categories of carriers by the sum power case (136-64). For idividual power costraits ad protocol P, the total rate is 318 bits per OFDM symbols duratio. The loss icurred compared to the sum power case ca be explaied
1 EURASIP Joural o Wireless Commuicatios ad Networig 35 P1P: rate versus power sum optimum /uiform LSR 1 LSD LRD 1 16 P1P: rate versus power sum optimum /uiform LSR 1 LSD LRD 1 3 14 5 1 Rate gai (%) 15 1 5 5 5 1 15 5 3 P t (dbw) Rate (bits) 1 8 6 4 5 1 15 5 3 P t (dbw) P1 P P1 opt P opt P1 uif P uif Figure 6: Rate gai with the optimized power allocatio compared to the uiform oe, versus P t (dbw) for the two protocols ad the sum power costrait. Taps of the λ sr () have a variace dbs above those associated with the λ sd () ad the λ rd (). Figure 7: Rate versus P t (dbw) for the two protocols ad uiform ad optimized power allocatio for the sum power costrait. Taps of the λ sr () have a variace 1 dbs above those associated with the λ sd () ad the λ rd (). i a maer idetical to that discussed for protocol P1. Ad agai the advatage of this protocol compared to P1 is visible. Systematic results have also bee produced for the two protocols, the sum power case, ad differet values of P t. For each value of P t the results reported are obtaied by averagig over 5 chael realizatios. The CIRs associated with the λ sr (), have a variace of dbs above those associated with the λ sd () ad the λ rd (). The results obtaied with the optimized power allocatio are cotrasted agaist uiform power allocatio. For protocol P1 with uiform power allocatio, the carrier allocatio to sets S s ad S r is performed as i the optimized case. The power available is uiformly divided betwee the N t carriers. For the carriers to be relayed, the per carrier power is further split betwee source ad relay accordig to the ratio associated with the saturatio of the decodability costrait (7). For protocol P, the allocatio of the carrier to set S s or S r is based o the compariso of λ sd (q) with λ sr (q) (α(q)/(1 α(q))). If N s carriers are allocated to set S s ad N t N s to set S r the total power is divided by N s N t N s = N t N s i order to tae ito accout the use of the two time slots for the carriers i S s. At this poit the reallocatio step is implemeted ad some carriers may be moved from S r to S s. For the carriers remaiig i set S r the power is further split amog source ad relay accordig to the ratio associated with the saturatio of the decodability costrait (7). Figure 5 reports the rate obtaied with the two protocols, ad for each protocol, with the optimized ad the uiform power allocatio. I order to have a better uderstadig of the gai associated with the optimized power allocatio with respect to the uiform oe, the rate gai i % betwee uiform power allocatio Rate gai (%) 4 35 3 5 15 1 5 P1P: rate versus power sum optimum /uiform LSR 1 LSD LRD 1 5 5 1 15 5 3 P t (dbw) P1 P Figure 8: Rate gai with the optimized power allocatio compared to the uiform oe, versus P t (dbw) for the two protocols ad the sum power costrait. Taps of the λ sr () have a variace 1 dbs above those associated with the λ sd () ad the λ rd (). ad optimized allocatio is also reported i Figure 6. The rate results (Figure 5) clearly show the higher efficiecy of protocol P compared to P1. This is due to the better use of the secod time slot for the orelayed carriers. For high values of P t ad protocol P, all carriers will be allocated to set S s (because of the reallocatio step). Because each carrier
EURASIP Joural o Wireless Commuicatios ad Networig 11 is used over the two time slots, the rate grows with a slope N t for P whereas the slope is oly N t with P1. The rate gai results (Figure 6) show how the rate gai evolves with P t. Clearly ad as expected, the beefit of the optimized power allocatio decreases with P t. For high values of P t the optimized power allocatio teds to become a uiform oe. Figures 7 ad 8 report similar results for the case where the CIRs associated with the λ sr (),haveavariaceof 1 dbs (istead of dbs) above those associated with the λ sd () ad the λ rd (). These results lead to similar coclusios. 6. Coclusio I this paper we cosidered a OFDM poit to poit li ehaced by meas of a relay. Whe a symbol is received by the relay o a certai toe, it may be relayed to the destiatio o the same toe. We have ivestigated the problem of power allocatio to the source ad to the relay i order to maximize the rate of the whole trasmissio for a global power costrait ad for idividual power costraits at the source ad at the relay. Two protocols have bee cosidered; the secod oe maes a better use of the secod time slot wheever the relay is iactive. It is assumed that the destiatio implemets MRC betwee what is received from the source ad what is received from the relay, for each toe. The DF operatig mode of the relay puts a additioal costrait o the desig. The carrier classificatio (whether a carrier has to be relayed or ot) has first bee ivestigated for the sum power case. The power allocatio problem has bee show to be of the waterfillig type with a specific costructio of the cotaier. It has also bee show how the problem for idividual costraits could be recast ito a equivalet waterfillig problem by usig the techique of equivalet powers ad equivalet chaels. It has bee proposed to fid iteratively the two Lagrage multipliers i this secod case by meas of a Newto-Raphso method implemeted for each possible carrier assigmet. Numerical results have bee provided to illustrate the schemes ad have show the advatage of protocol P over protocol P1. Future wor will be devoted to the cases of multiple relays, operfect chael state iformatio ad a refiemet of the power allocatio across the two sigalig itervals. Moreover, codig will also be icluded i the trasmissio scheme ad tae ito accout. Besides these topics, the (pea to average power ratio) PAPR might also be a problem to be cosidered. PAPR issues are well ow with OFDM trasmissio ad are liely to be impacted by power allocatio. Acowledgmets The authors would lie to tha the Walloo Regio DGTRE Naotic-COSMOS project, the FP6 project COOPCOM ad the FP7 Networ of Excellece NEWCOM for their fiacialsupport.partsofthisworhavebeereportedi IEEE SCVT 7, IEEE ICC 8 ad ISWPC 8. Refereces 1 A. Sedoaris, E. Erip, ad B. Aazhag, Icreasig upli capacity via user cooperative diversity, i Proceedigs of the IEEE Iteratioal Symposium o Iformatio Theory, p. 156, August 1998. A. Sedoaris, E. Erip, ad B. Aazhag, User cooperatio diversity part I: system descriptio, IEEE Trasactios o Commuicatios, vol. 51, o. 11, pp. 197 1938, 3. 3 A. Sedoaris, E. Erip, ad B. Aazhag, User cooperatio diversity part II: implemetatio aspects ad performace aalysis, IEEE Trasactios o Commuicatios, vol. 51, o. 11, pp. 1939 1948, 3. 4 T. M. Cover ad A. A. El-Gamal, Capacity theorems for the relay chael, IEEE Trasactios o Iformatio Theory, vol. 5, o. 5, pp. 57 584, 1979. 5 G. Kramer, M. Gastpar, ad P. Gupta, Cooperative strategies ad capacity theorems for relay etwors, IEEE Trasactios o Iformatio Theory, vol. 51, o. 9, pp. 337 363, 5. 6 T. C.-Y. Ng ad W. Yu, Joit optimizatio of relay strategies ad resource allocatios i cooperative cellular etwors, IEEE Joural o Selected Areas i Commuicatios, vol. 5, o., pp. 38 339, 7. 7 I. Hammerstrom ad A. Wittebe, O the optimal power allocatio for oregeerative OFDM relay lis, i Proceedigs of the IEEE Iteratioal Coferece o Commuicatios, vol. 1, pp. 4463 4468, 6. 8 W. Yig, Q. Xi-Chu, W. Tog, ad L. Bao-Lig, Power allocatio ad subcarrier pairig algorithm for regeerative OFDM relay system, i Proceedigs of the 65th IEEE Vehicular Techoy Coferece (VTC 7), pp. 77 731, 7. 9 B. Gui ad L. J. Cimii Jr., Bit loadig algorithms for cooperative OFDM systems, i Proceedigs of the IEEE Military Commuicatios Coferece (MILCOM 7), pp. 1 7, October 7. 1 Y. Ma, N. Yi, ad R. Tafazolli, Bit ad power loadig for OFDM-based three-ode relayig commuicatios, IEEE Trasactios o Sigal Processig, vol. 56, o. 7, pp. 336 347, 8. 11 T. Sarteaer, J. Louveaux, ad L. Vadedorpe, Balaced capacity of wirelie multiple access chaels with idividual power costraits, IEEE Trasactios o Commuicatios, vol. 56, o. 6, pp. 95 936, 8. 1 R. S. Cheg ad S. Verdu, Gaussia multiaccess chaels with ISI: capacity regio ad multiuser water-fillig, IEEE Trasactios o Iformatio Theory, vol. 39, o. 3, pp. 773 785, 1993. 13 J. N. Laema, D. N. C. Tse, ad G. W. Worell, Cooperative diversity i wireless etwors: efficiet protocols ad outage behaviour, IEEE Trasactios o Iformatio Theory, vol. 5, pp. 36 38, 4. 14 S. Boyd ad L. Vadeberghe, Covex Optimizatio, Cambridge Uiversity Press, Cambridge, UK, 4. 15 J. Louveaux, R. Torrea, ad L. Vadedorpe, Efficiet algorithm for optimal power allocatio i OFDM trasmissio with relayig, i Proceedigs of the IEEE Iteratioal Coferece o Acoustics, Speech, ad Sigal Processig (ICASSP 8), pp. 357 36, Las Vegas, Calif, USA, May 8.