Energy-Efficient Link Adaptation in Frequency-Selective Channels

Transcription

1 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 58, NO., FEBUAY Energy-Effcent Lnk Adaptaton n Frequency-Selectve Channels Guowang Mao, Student Member, IEEE, Nageen Hmayat, Member, IEEE, and Geoffrey Ye L, Fellow, IEEE Abstract Energy effcency s becomng ncreasngly mportant for small form factor moble devces, as battery technology has not kept up wth the growng requrements stemmng from ubqutous multmeda applcatons. Ths paper adesses lnk adaptve transmsson for maxmzng energy effcency, as measured by the throughput per Joule metrc. In contrast to the exstng water-fllng power allocaton schemes that maxmze throughput subject to a fxed overall transmt power constrant, our scheme maxmzes energy effcency by adaptng both overall transmt power and ts allocaton, accordng to the channel states and the crcut power consumed. We demonstrate the exstence of a unque globally optmal lnk adaptaton soluton and develop teratve algorthms to obtan t. We further consder the specal case of flat-fadng channels to develop an upper bound on energy effcency and to characterze ts varaton wth bandwdth, channel gan and crcut power. Our results for OFDM systems demonstrate mproved energy savngs wth energy optmal lnk adaptaton as well as llustrate the fundamental tradeoff between energy-effcent and spectrum-effcent transmsson. Index Terms Energy effcency, lnk adaptaton, frequency selectve channel, energy aware, OFDM. I. INTODUCTION THE qualty of wreless channel vares wth tme and frequency. Therefore, lnk adaptaton can be used to mprove transmsson performance. Wth lnk adaptaton, modulaton order, codng rate, and transmt power can be selected accordng to channel state nformaton (CSI). Earler research on lnk adaptaton focuses on power allocaton to mprove channel capacty subject to overall power constrant. Optmal power allocaton for frequency-selectve channels has been nvestgated n []. The termnology, adaptve modulaton, was frst used n [] even though work on adaptve modulaton [3] had been reported before. In addton to throughput mprovement, energy effcency s becomng ncreasngly mportant for moble communcatons because of the slow progress of battery technology [4] and growng requrements of anytme and anywhere multmeda applcatons. Wth suffcent battery power, lnk adaptaton can be geared toward peak performance delvery. However, wth lmted battery capacty, lnk adaptaton could be adapted Paper approved by E. Serpedn, the Edtor for Synchronzaton and Sensor Networks of the IEEE Communcatons Socety. Manuscrpt receved November 3, 008; revsed Aprl 30, 009. G. Mao and G. Y. L are wth the School of Electrcal and Computer Engneerng, Georga Insttute of Technology, Atlanta, GA (e-mal: gmao3@gatech.edu, lye@ece.gatech.edu). N. Hmayat s wth the Communcatons Technology Lab., Intel Corporaton, Santa Clara, CA (e-mal: nageen.hmayat@ntel.com). Ths work was supported by Intel Corp. and the U.S. Army esearch Laboratory under the Collaboratve Technology Allance Program, Cooperatve Agreement DAAD Dgtal Object Identfer 0.09/TCOMM /0$5.00 c 00 IEEE toward energy conservaton to mnmze battery an. Energyeffcent communcaton also has the desrable beneft of reducng nterference to other co-channel users as well as lessenng envronmental mpacts, e.g. heat dsspaton and electronc polluton. Hence, recent research has focused on energy-effcent lnk adaptaton technques [5] [7]. It s shown n [5] that when the transmsson bandwdth approaches nfnty, the mnmum receved sgnal energy per bt for relable communcaton over addtve whte Gaussan nose (AWGN) channels, approaches.59 db. For band-lmted transmsson, the lowest order modulaton should be used [6]. However, the nvestgaton n [5], [6] does not account for addtonal crcut power consumed durng transmsson. Energy dsspaton of both transmtter crcuts and radofrequency output s nvestgated n [8], where the modulaton level s adapted to mnmze the energy consumpton accordng to the smulaton observatons. In [7], these deas are extended to a detaled analyss of crcut and transmt powers for both adaptve multple quaature ampltude modulaton (M- QAM) and multple frequency shft keyng (MFSK) n AWGN channels for short range energy-effcent communcatons. Orthogonal frequency dvson multplexng (OFDM) has emerged as a prmary modulaton scheme for the nextgeneraton broadband wreless standards [9], [0]. The power allocaton and bt-loadng algorthms for OFDM are summarzed n Chapter 3 of []. Whle extensve research has been conducted to mprove throughput [], [], lmted work has been done to adess the energy-effcent communcaton for OFDM systems. In ths paper, we adess the energy-effcent lnk adaptaton for frequency-selectve fadng channels. We account for both crcut and transmt powers when desgnng lnk adaptaton schemes and emphasze energy effcency over peak rates or throughput. The proposed lnk adaptaton balances crcut power consumpton and transmsson power to acheve the maxmum energy effcency, whch s defned as the number of bts transmtted per Joule of energy. In contrast to the exstng water-fllng power allocaton schemes that maxmze throughput subject to overall transmt power constrants, our scheme adapts both the overall transmt power and ts allocaton accordng to the states of all subchannels and the crcut power consumpton to maxmze the energy effcency. We demonstrate the exstence of a unque globally optmal lnk adaptaton soluton and provde teratve algorthms to obtan ths optmum. Whle the usefulness of our technque s llustrated usng frequency selectve OFDM as an example n ths paper, the soluton developed s applcable to more general transmsson scenaros where transmsson occurs over resources experencng dfferent channel condtons.

2 546 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 58, NO., FEBUAY 00 The rest of the paper s organzed as follows. In Secton III, we nvestgate optmal condtons for energy-effcent transmsson and develop algorthms to obtan the globally optmal soluton. In Secton III-B, we consder a specal case when the channel s wth flat fadng. We also consder energyeffcent lnk adaptaton when the user has ether data rate requrement or peak power lmt n Secton IV. As an example of energy-effcent lnk adaptaton, we apply the energy-effcent scheme n OFDM systems and provde smulaton results to demonstrate energy effcency mprovement n Secton VI. Fnally, we conclude the paper n Secton VII. II. POBLEM FOMULATION In ths secton, we formulate the problem of energy-effcent lnk adaptaton. Assume that K subchannels are used for transmsson, each wth a dfferent channel gan. An example of ths scenaro s OFDM transmsson over frequency-selectve channels. Assume block fadng [3], [4], that s, the channel state remans constant durng each data frame and s ndependent from one to another. Denote the data rate on Subchannel as r and the data rate vector on all subchannels as =[r,r,,r K ] T, () where [] T s the transpose of a vector. The data rate vector,, depends on the channel state, codng, and power allocaton. Correspondngly, the overall data rate s = K r. () = For a gven channel state, the transmt power on each subchannel s determned by the requrement of relable data transmsson. If we denote W as the subchannel bandwdth, N o the power spectral densty, g the power gan, and P T the allocated transmt power on Subchannel, the channel output sgnal-to-nose rato (SN) wll be η = P T g (3) N o W and the achevable data transmsson rate r s determned by [5] r = W log( + η ), (4) Γ where Γ s the SN gap that defnes the gap between the channel capacty and a practcal codng and modulaton scheme. The SN gap depends on the codng and modulaton scheme used and on the target probablty of error. For a coded quaature ampltude modulaton (QAM) system, the gap s gven by [5] Γ=9.8+γ m γ c (db), (5) where γ m s the system desgn margn and γ c s the codng gan. For Shannon capacty [6], Γ=0dB. Denote the overall transmt power as P T () and K = P T () = P K T = ζ = (e r W N o W Γ ), (6) g ζ where ζ [0, ] s the power amplfer effcency and depends on the desgn and mplementaton of the transmtter. P T () s strctly convex and monotoncally ncreasng n. In fact, the developed theory and approaches can be used for any P T () that s strctly convex and monotoncally ncreasng n wth P T (0) =0,where0 =[0, 0,, 0] T. In addton to transmt power, moble devces also ncur addtonal crcut power durng transmssons whch s relatvely ndependent of the transmsson rate [8], [7]. Whle the transmt power models all the power used for relable data transmsson, we let the crcut power represents the average energy consumpton of devce electroncs, such as mxers, flters, and dgtal-to-analog converters, and ths porton of energy consumpton excludes that of the power amplfer and s ndependent of the transmsson state. If we denote the crcut power as P C, the overall power consumpton gven a data rate vector wll be P () =P C + P T (). (7) For energy-effcent communcatons, t s desrable to maxmze the amount of data sent wth a gven amount of energy. Hence, gven any amount of energy e consumed n a duraton, t,.e. e = t(p C + P T ()), the moble wants to send a maxmum amount of data by choosng the data rate vector to maxmze t e, (8) whch s equvalent to maxmzng U() = e/ t = P C + P T (). (9) U() s called energy effcency. The unt of the energy effcency s bts per Joule, whch has been frequently used n lterature for energy-effcent communcatons [5], [6], [8] [0]. The optmal energy-effcent lnk adaptaton acheves maxmum energy effcency,.e. =argmax U() = arg max P C + P T (). (0) Note that f we fx the overall transmt power, the objectve of Equaton (0) s equvalent to maxmzng the overall throughput and the exstng water-fllng power allocaton approach [] gves the soluton. However, besdes adaptng the power dstrbutons on all subchannels, the overall transmt power can also be adapted accordng to the states of all subchannels to maxmze the energy effcency. Hence, the soluton to Equaton (0) s n general dfferent from exstng power allocaton schemes that maxmze throughput wth power constrants. III. PINCIPLES OF ENEGY-EFFICIENT LINK ADAPTATION In the followng, we demonstrate that a unque globally optmal data rate vector always exsts and gve the necessary and suffcent condtons for a data rate vector to be globally optmal. A. Condtons of Optmalty The concept of quasconcavty wll be used n our dscusson and s defned as [].

3 MIAO et al.: ENEGY-EFFICIENT LINK ADAPTATION IN FEQUENCY-SELECTIVE CHANNELS 547 Defnton. A functon f, whch maps from a convex set of real n-dmensonal vectors, D, to a real number, s called strctly quasconcave f for any x, x Dand x = x, f(λx +( λ)x ) > mn{f(x ),f(x )}, () for any 0 <λ<. Any strctly monotonc functon s quasconcave. Besdes, any strctly concave functon s also strctly quasconcave but the reverse s not generally true. An example s the Gaussan functon, whch s strctly quasconcave but not concave. It s proved n Appendx A that U() has the followng propertes. Lemma. If P T () s strctly convex n, U() s strctly quasconcave. Furthermore, U() s ether strctly decreasng or frst strctly ncreasng and then strctly decreasng n any r of,.e. the local maxmum of U() for each r exsts at ether 0 or a postve fnte value. For strctly quasconcave functons, f a local maxmum exsts, t s also globally optmal []. Hence, a unque globally optmal transmsson rate vector always exsts and ts characterstcs are summarzed n Theorem accordng to the proofs n Appendx A. Theorem. If P T () s strctly convex, there exsts a unque globally optmal transmsson data rate vector = [r,r,,r K ]T for (0), where r s gven by () when PC+PT ((0) ) (0) 0,.e. P T ( ) r () when PC +PT ((0) ) < (0) PT () = r (0) = P C+P T ( ) = U( ); PT () = r (0), U() r, r =0, = where (0) =[r,r,,r, 0,r +,,r K ] and (0) = j = r j,.e. the overall data rate on all other subchannels except. Theorem has clear physcal nsghts. P C +P T ( (0) ) s the power consumpton of both crcut and all other subchannels P C +P T ( (0) ) when Subchannel s not used. s the per-bt (0) energy consumpton when Subchannel s not used and the overall per-bt energy consumpton needs to be mnmzed for energy-effcent communcatons. s the P T () r = (0) per-bt energy consumpton transmttng nfntely small data rate on Subchannel condtoned on the optmal status of all other subchannels. Hence, Subchannel should not transmt anythng when PC +PT ((0) ) (0) < PT () r = (0) =.Otherwse, there should be a tradeoff between the desred data rate on Subchannel and the ncurred power consumpton. The tradeoff closely depends on the power consumpton of both crcuts and transmsson on all other subchannels and can be found through the unque zero dervatve of U() wth respect to r. To further understand Theorem, we consder an example when each subchannel acheves the Shannon capacty and the transmt power on each subchannel s gven n (6) wth Γ=0 db and ζ =. The overall transmt power s K P T () = (e r k N o W W ). () g k k= Accordng to Condton () of Theorem, when r k > 0, we have = = U( ). (3) P T () r k e r k N W o g k Hence, the transmt power on Subchannel k s P Tn =(e r k N o W W ) = W g k U( ) N ow, (4) g k W whch s a water-fllng to level U( ). Snce the water level s determned by the optmal energy effcency, we refer to our scheme as dynamc energy-effcent water-fllng. Note that whle the absolute value of power allocaton s determned by the maxmum energy effcency U( ), whch reles on both the crcut power and channel state, the relatve dfferences of power allocatons on dfferent subchannels depend only on the channel gans on those subchannels. B. A Specal Case: When the Channel s Flat Fadng To facltate the understandng of the fundamental dependence of energy effcency on the channel gan, crcut power, and bandwdth, we consder a specal case that the channel s experencng flat fadng n ths secton. Hence, all subchannels are wth the same channel gan and the same lnk adaptaton s appled on all subchannels. The overall data rate s = Kr. (5) Accordng to Theorem, the optmal transmsson data rate follows mmedately and s summarzed by Theorem, where the upper bound s proved n Appendx B. Theorem. If P T () s monotoncally ncreasng and strctly convex n, there exsts a unque globally optmal transmsson data rate to maxmze energy effcency and s gven by = P C + P T ( ), (6) P T ( ) where P T ( ) s the frst order dervatve of functon P T ( ). Besdes, energy effcency s upper bounded by P (0). T When Shannon capacty s acheved n AWGN channels, g the upper bound s N o. In the followng, we nvestgate some basc propertes of energy-effcent lnk adaptaton. Propostons,, and 3 summarze the mpact of channel gan, crcut power, and the number of subchannels on the optmal energy-effcent transmsson, and are proved n Appendx C. Proposton. Both the data rate and energy effcency ncrease wth channel gan. Proposton. The data rate ncreases wth crcut power, whle the energy effcency decreases wth t. Wth zero crcut power, the hghest energy effcency, P T (0), s obtaned by transmttng wth nfnte small data rate.

4 548 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 58, NO., FEBUAY 00 From Proposton, when crcut power domnates power consumpton, whch s usually true wth short-range communcaton, the hghest data rate should be used to fnsh transmsson as soon as possble, whch has been commonly assumed by most MAC layer energy-effcent optmzaton schemes as descrbe n the ntroducton of ths paper. However, when the crcut power s neglgble, whch s usually true wth long-range communcaton lke satellte communcatons, the lowest data rate should be used, whch concdes wth the results n [6] and []. Proposton 3. The data rate on each subchannel decreases wth ncreasng number of subchannels whle the energy effcency ncreases wth t. Wth nfnte number of subchannels, the hghest energy effcency, P T (0), s obtaned by transmttng wth nfnte small data rate. Propostons,, and 3 dscover three ways to mprove energy effcency: ncreasng channel power gan, reducng crcut power, and allocatng more subchannels. The energyeffcency upper bound s acheved by transmttng wth nfnte small data rate when ether crcut power s zero or nfnte number of subchannels s assgned. IV. CONSTAINED ENEGY-EFFICIENT LINK ADAPTATION In ths secton, we study energy-effcent lnk adaptaton when user has ether a data rate requrement or a peak power lmt. Wth a data rate requrement Γ, the energy-effcent lnk adaptaton s gven by subject to ˆ =argmax Γ. P C + P T (), (7a) (7b) If the optmal data rate vector wthout constrant n (0) satsfes Γ, t s also the soluton to Problem (7),.e. ˆ =. Otherwse, Problem (7) s equvalent to ˆ =argmax Γ subject to Γ P C + P T () =argmn P T (), (8a) =Γ. (8b) Snce P T () s strctly convex, a unque globally optmal ˆ exsts. Denote f k (r k )= P T () (9) r k and ts nverse functon to be f k (). Then ˆ can be easly obtaned va the Lagrangan technque[3] and s ˆr k =max { f k (λ), 0} (0) for k =,,K,whereλ s determned by K k= ˆr k =Γ. () When the channel capacty s acheved on each subchannel, the correspondng optmal power allocaton s a water-fllng allocaton, whch acheves the sum channel capacty Γ. Smlarly, wth a maxmum transmt power constrant, the problem s to fnd subject to =argmax P T () P m. P C + P T (), (a) (b) If the optmal data rate vector wthout constrant n (0) satsfes P T ( ) P m, t s also the soluton to Problem (),.e. =. Otherwse, va the the Lagrangan technque agan, we have the unque optmal soluton as follows r k =max{ f k (λ), 0},k =,,K, (3) where λ s determned by P T ( )=P m. (4) When channel capacty s acheved on each subchannel, the power allocaton s the classcal water-fllng where the water level s determned by P m []. V. ALGOITHM DESIGN Theorem provdes the necessary and suffcent condtons for a rate vector to be the unque and globally optmum one. However, t s usually dffcult to drectly solve the jont nonlnear equatons accordng to Theorem to obtan the optmal vector. Therefore, we develop teratve methods to search the optmal for maxmzng U(). The global optmalty of the proposed methods s guaranteed by the strct quasconcavty of U(). In the followng, we descrbe our low-complexty teratve algorthms. A. Gradent Asssted Bnary Search When there s only one subchannel, Lemma shows that functon U(r) has a unque r such that for any r < r, du(r) > 0, andforanyr>r du(r), < 0. Hence, we have the followng lemma to seek two ponts r and r such that r r r. Proposton 4. Let the ntal settng r [0] > 0 and set α>. For any 0, let { r [] du(r) r [+] = α < 0 r [0]. (5) αr [] otherwse epeat (5) untl r [I] such that du(r) has a dfferent sgn r [I] from du(r) Thenr r [0]. must be between r [I] and r [I ]. To locate r between r and r,letˆr = r+r.if du(r) = ˆr 0, r s found. If du(r) < 0, r <r ˆr < ˆr and replace r wth ˆr; otherwse, replace r wth ˆr. Thsleadstothegradent asssted bnary search (GABS) for maxmzng U(r), whch s summarzed n Table I. B. Bnary Search Asssted Ascent To fnd the optmal data rate vector for the multple subchannel case, we desgn a gradent ascent method to produce

5 MIAO et al.: ENEGY-EFFICIENT LINK ADAPTATION IN FEQUENCY-SELECTIVE CHANNELS 549 Ensemble average of normalzed energy effcency K=4 K=6 K=64 K=8 K=5 K=04 Probablty dstrbuton functon K=4 K=6 K=64 K=8 K=5 K= Number of teratons (a) Convergng process: relatonshp between ensemble average and teratons Number of teratons (b) Probablty dstrbuton functon of the number of teratons for convergence Fg. : Convergence rate of BSAA. TABLE I: Gradent asssted bnary search Algorthm GABS(r o ) ( algorthm for sngle-subchannel transmsson. ) Input: ntal guess: r o > 0 Output: optmal transmsson rate: r. r = r o, h du(r), ntalze α> (e.g.0) r. f h < 0 ( seek r and r such that r <r <r ) 3. then r r, r r α,andh du(r) r 4. whle h < 0 5. do r r, r r α,andh du(r) r 6. else r r α and h du(r) r 7. whle h > 0 8. do r r, r r α, andh du(r) 9. whle no convergence ( seek r between r and r ) 0. do ˆr r+r ; ĥ du(r) ˆr. f ĥ>0. then r = ˆr; 3. else r = ˆr 4. return ˆr a maxmzng sequence [], n =0,,,and [ + [+] = [] + μ U( )] [], (6) where [] + sets the negatve part of the vector to be zero, μ>0 s the search step sze, and U( [] ) s the gradent at teraton. Wth suffcently small step sze, U( [+] ) wll be always bgger than U( [] ) except when U( [] )=0 that ndcates the optmalty of [] [3]. However, small step sze leads to slow convergence. Besdes, each element of the gradent depends on the correspondng subchannel power gan, whch potentally dffers from each other by orders of r magntude. Hence, a lne search of the optmal step sze needs to cover a large range to assure global convergence on all subchannels, whch s computatonally expensve. Therefore, at each [], an effcent algorthm s needed to fnd the optmal step sze. Denote [ +). f (μ) =U( [] + μ U( )] [] (7) Smlar to the proof of Lemma, t s easy to show that g (μ) s also strctly quasconcave n μ and has a unque globally maxmum μ such that for any μ<μ df, (μ) dμ > 0, andfor any μ>μ df, (μ) dμ < 0. Let U( [] )=[ˆg, ˆg,, ˆg K ]. n GABS to be eplace du(r) df (μ) dμ =[ U([+] )] T G[], (8) where G[] = d[[] +μ U( [] )] + dμ =[ g, g,, g K ],nwhch g k = ˆg k f the kth component of [] +μ U( [] ) s postve and g k =0otherwse. Then GABS can be used for quck locaton of the optmal step sze. Ths leads to the bnary search asssted ascent (BSAA) algorthm n Table II. TABLE II: Bnary search asssted ascent Algorthm BSAA( o ) ( algorthm for mult-subchannel transmsson. ) Input: ntal guess: o (default transmsson rate can be used) Output: optmal transmsson rate vector:. = o,. whle no convergence 3. do use GABS to fnd the optmal step sze μ ; 4. =[ + μ U()] + 5. return C. The ate of Convergence Whle the global convergence of both GABS and BSAA s guaranteed by the strct quasconcavty of U() [4], we

6 550 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 58, NO., FEBUAY 00 further study the convergence rate n ths secton. Theorem 3 characterzes the convergence of GABS and s proved n Appendx D. Theorem 3. GABS converges to the globally optmal transmsson data rate r.arater, whch satsfes r r ε, can be found wthn at most M teratons, where M s the mnmum nteger such that M log ( (α )r ε ). It s dffcult to theoretcally analyze the global convergence rate of BSAA because of the nonconcavty of U(). Instead, we run numercal smulatons and observe the convergence. Fgure (a) llustrates the mprovement of energy effcency wth teratons. Here we assume the channel gan of each subchannel has aylegh dstrbuton wth a unt average. The crcut power s 5. The nose power on each subchannel s 0.0. The transmt power s gven by Equaton (6) wth Γ=0 db. The energy effcency s normalzed by the optmal value and the curves are the ensemble averages of 5000 channel nstances. Fgure (b) shows the correspondng probablty dstrbuton functons of the numbers of teratons necessary for convergence. In both fgures, we vary the number of subchannels to verfy ts mpact on the convergence rate. We can see that BSAA converges very fast to the global optmum, even wth 04 subchannels. VI. SIMULATION ESULTS FO OFDM The proposed energy-effcent lnk adaptaton s general and can be appled to dfferent knds of OFDM, MIMO, and MIMO-OFDM systems. To apply t, we only need to fnd the transmt power relatonshp P T () of those systems. In ths secton, we dscuss the optmal energy-effcent lnk adaptaton for OFDM wth subchannelzaton as an example. A. Modelng of OFDM wth Subchannelzaton In OFDM systems wth subchannelzaton, subcarrers are grouped nto subchannels and the subcarrers formng one subchannel may, but not necessarly be adjacent, such as the contguous and dstrbuted subchannelzaton schemes n 80.6e [9]. Each subchannel s treated to be flat fadng and the effectve channel power gan, g, rather than physcal channel power gan of each subcarrer, s used as a metrc. For smplcty, g s the average of channel power gans of all subcarrers wthn the subchannel. Note that classcal OFDM s a specal case when each subchannel has one subcarrer. The frame structure s shown n Fgure. Each transmsson slot conssts of a data nterval, T s, and a sgnallng nterval, τ. In each data nterval, l symbols are transmtted. We use uncoded M-QAM. The transmt power on each subchannel needs to be determned. Consder Subchannel that conssts of c subcarrers. The number of bts transmtted per symbols on each subcarrer s b = r c (T s+τ ) l. Hence, the modulaton order M s gven by M = b = Br,whereB = (Ts+τ ) c l.thebt-error rate (BE) for coherently detected M-QAM wth Gray mappng over an AWGN channel s approxmated by [5] ( P e (γ) 0.exp.5γ ), (9) M Data Interval n 3 T s l- Sgnalng Interval Fg. : Frame structure l τ where γ s the sgnal-to-nose rato (SN). For a BE target, P e, the requred SN on Subchannel s γ = 3 ( M )ln(5p e )= 3 ( Br )ln(5p e ). (30) Hence, the overall transmt power on Subchannel s P T (r )= γ c N o W = A ( Br ), (3) g where W s the sgnal bandwdth of each subcarrer and A = c ln(5p e )N o W. (3) 3g Assumng no couplng between transmt powers among subchannels, the overall transmt power wll be the cumulatve of the transmt powers of all subchannels, that s, P T () = K P T (r ), (33) = whch s monotoncally ncreasng and strctly convex n. The energy-effcent lnk adaptaton mmedately follows from Secton III. B. Performance Comparson In ths secton, we compare the performance of energyeffcent OFDM transmsson wth that of tradtonal transmsson schemes. The system parameters are lsted n Table III. The Internatonal Telecommuncaton Unon (ITU) pedestran channel model B [6] s used to mplement the multpath frequency-selectve fadng. We mplement two subchannelzaton schemes as n Fgure 3, fxed-nterval and contguous, both of whch group 0 subcarrers nto a subchannel. In the fxed-nterval subchannelzaton, one aws subcarrers out of all subcarrers wth a fxed nterval to form a subchannel, whle n the contguous one, each subchannel conssts of a block of contguous subcarrers. Fgures 4(a) and 4(b) and Fgures 5(a) and 5(b) compare energy effcency and throughput of dfferent transmsson schemes wth contguous subchannelzaton and wth fxednterval subchannelzaton respectvely. Two energy-effcent OFDM transmsson schemes are mplemented: FS EE, that s the optmal energy-effcent transmsson developed n ths paper, and flat EE, that treats the channel as flat fadng. Transmssons wth both fxed and adaptve QAM modulatons are also mplemented for comparson. For fxed modulaton, the transmt power s adapted to meet BE requrement

7 MIAO et al.: ENEGY-EFFICIENT LINK ADAPTATION IN FEQUENCY-SELECTIVE CHANNELS 55 TABLE III: System parameters Carrer frequency.5 GHz Subcarrer number 56 Subcarrer bandwdth 0 khz BE requrement 0 3 bol number of data nterval, l 00 Tme duraton of data nterval, T s 0.0s Tme duraton of sgnallng nterval, τ 0.00s Thermal nose power, N o -4 dbw/mhz User antenna heght.6 m BS antenna heght 40 m Envronment Macro cell n urban area Crcut power, P C 00 mw Modulaton Uncoded M-QAM Subchannelzaton Fxed-nterval and contguous Propagaton Model Okumura-Hata model Shadowng Log-normal wth standard devaton of 0 db Frequency-selectve fadng ITU pedestran channel B User speed 3 km/h Fxed-nterval Subchannelzaton Subchannel Subchannel subchannels. Ths ndcates energy-effcent lnk adaptaton treatng channels to be flat fadng s suffcent for performance optmzaton. Contguous Subchannelzaton K+ K+ Subcarrer ndex Subchannel Subchannel C+ C C+ Subcarrer ndex Fg. 3: OFDM subchannelzaton (K subchannels, each wth c subcarrers) whle not exceedng 5 dbm maxmum power constrant. For adaptve modulaton, transmt power s equally dstrbuted over all subchannels and the modulaton s adapted to meet BE requrement. From Fgures 4(a) and 4(b), fxed and adaptve modulatons perform closely to each other, especally when far away from BS, for both energy effcency and throughput, when the maxmum transmt power s 5 dbm. By ncreasng the transmt power from 5 dbm to 5 dbm, the throughput of adaptve modulaton ncreases, however, the energy effcency frst ncreased and then decreases. Due to the global optmalty, the proposed energy-effcent transmsson for frequency-selectve channels always acheves the hghest energy effcency, and outperforms the others by at least 5%. However, the throughput s not necessarly maxmum; the other schemes, especally the adaptve QAM modulaton wth 5 dbm transmt power, sacrfce power to obtan hgher throughput. Smlar results can also be observed n Fgures 5(a) and 5(b). Furthermore, we note that when fxed-nterval subchannelzaton s used, dfferent subchannels have trval dfferences n average channel gan and the energy-effcent transmsson treatng the channel to be flat fadng performs the same as the one consderng the dfference of dfferent VII. CONCLUSION In ths paper, we have nvestgated the energy-effcent lnk adaptaton. Whle the usefulness of the proposed technque s llustrated usng frequency-selectve OFDM as an example, the soluton developed s applcable to more general transmsson scenaros where transmsson occurs over resources experencng dfferent channel condtons. Jont crcut and transmt power consumptons are taken nto account to maxmze energy effcency rather than throughput. We demonstrate the exstence of a unque globally optmal lnk adaptaton soluton and provde teratve algorthms to obtan ths optmum. The optmal power allocaton s shown to be a dynamc waterfllng where the water level s determned by the maxmum energy effcency. We further consder a specal case when the channel s experencng flat fadng and show the upper bound of energy effcency as well as two ways to acheve ths bound. We explctly demonstrate that energy effcency s mproved by ncreasng channel power gan, bandwdth, and by reducng crcut power consumpton. From the smulaton results, we observed at least 5% mprovement n energy utlzaton when frequency selectvty s exploted and the mprovement depends on how much frequency dversty exsts wthn the channels. APPENDIX A POOF OF LEMMA Proof: Denote the upper contour sets of U() as S α = { ર 0U() α}, (A.34) where symbol ર denotes vector nequalty and ર 0 means each element of s nonnegatve. Accordng to Proposton

8 55 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 58, NO., FEBUAY 00 Energy effcency (kbts/joule) FS EE Flat EE Fxed QAM, Adapt Power, 5dBm Adapt QAM, Fxed Power, 5dBm Adapt QAM, Fxed Power, 0dBm Adapt QAM, Fxed Power, 5dBm Throughput (Mbts/s) FS EE Flat EE Fxed QAM, Adapt Power, 5dBm Adapt QAM, Fxed Power, 5dBm Adapt QAM, Fxed Power, 0dBm Adapt QAM, Fxed Power, 5dBm Dstance to BS (km) (a) Energy effcency Dstance to BS (km) (b) Throughput Fg. 4: Performance comparson for contguous subchannelzaton. Energy effcency (kbts/joule) FS EE Flat EE Fxed QAM, Adapt Power, 5dBm Adapt QAM, Fxed Power, 5dBm Adapt QAM, Fxed Power, 0dBm Adapt QAM, Fxed Power, 5dBm Throughput (Mbts/s) FS EE Flat EE Fxed QAM, Adapt Power, 5dBm Adapt QAM, Fxed Power, 5dBm Adapt QAM, Fxed Power, 0dBm Adapt QAM, Fxed Power, 5dBm Dstance to BS (km) (a) Energy effcency Dstance to BS (km) (b) Throughput Fg. 5: Performance comparson for fxed-nterval subchannelzaton. C.9 of [], U() s strctly quasconcave f and only f S α s strctly convex for any real number α. Whenα<0, no ponts exst on the contour U() =α. Whenα =0, only 0 s on the contour U(0) =α. Hence, S α s strctly convex when α 0. Now we nvestgate the case when α>0. S α s equvalent to S α = { ર 0αP C + αp T () 0}. Snce P T () s strctly convex n, S α s also strctly convex. Hence, we have the strct quasconcavty of U(). The partal dervatve of U() wth r s U() r = P C + P T () PT () (P C + P T ()) β(r ) (P C + P T ()), (A.35) where P T () s the frst partal dervatve of P T () wth respect to r. Accordng to Lemma, f r exsts such that U() r =0, t s unque,.e. f there s a r r=r such that β(r )=0, t s unque. In the followng, we nvestgate the condtons when r exsts. The dervatve of β(r ) s β (r )= P T () < 0, (A.36) where P T () s the second partal dervatve of P T () wth respect to r. Hence, β(r ) s strctly decreasng. Accordng to the L Hoptal s rule, t s easy to show that lm β(r ) = lm (P C + P T () P T ()) r > r > ( ) P C + P T () P T = lm () r r > = lm r > r ) (P T () P T () P T () r = lm P r T ()r < 0. > (A.37)

9 MIAO et al.: ENEGY-EFFICIENT LINK ADAPTATION IN FEQUENCY-SELECTIVE CHANNELS 553 Besdes, lm β(r ) = lm (P C + P T () P T ()) r >0 r >0 = P C + P T ( (0) ) (0) P T ((0) ), (A.38) where (0) = [r,r,,r, 0,r +,,r K ] T and (0) = j = r j. ( o )When P C + P T ( (0) ) (0) P T ((0) ) 0, lm r >0 β(r ) 0. Together wth (A.37), we see that t exsts and U() s frst strctly ncreasng and then strctly decreasng n r. ( o )When P C + P T ( (0) ) (0) P T ((0) ) < 0, lm r >0 β(r ) < 0. Together wth (A.36) and (A.37), t does not exst. However, U() s always strctly decreasng n r. Hence, U() s maxmzed at r =0. Lemma s readly obtaned. APPENDIX B POOF OF THE UPPEBOUND IN THEOEM Proof: U() = P C+P T () P T (). Denote ˆU() = P T (). ˆU () = d ˆU() d = PT () P T () PT (). Accordng to the L Hoptal s rule, lm 0 ˆU () = lm 0 P T () P T () lm 0 P T () P T () = P P T ()P T = lm () T () 0 0. Besdes (P T ()) P T () P T () s 0 when =0and has negatve dervatve when >0. Hence, P T () P T () < 0 when >0. Thus, ˆU () s negatve when >0and ˆU() s maxmzed when approaches zero,.e. U() lm 0 P T (0). P T () = APPENDIX C POOF OF POPOSITIONS,, AND 3 Proof: Denote P (r) to be the receved power on a subchannel for relable detecton when the data rate on the subchannel s r. We have P T () = KP(r) where g s the channel power gan. It s easy to see that P (r) s monotoncally ncreasng and strctly convex, and P T (0) = P (0) = 0. Accordng to Theorem, we have P T ( ) = P C + P T ( ), whch s equvalent to P ( K ) KP ( K )=P Cg. By dfferentatng the left ( P ( K g = KP( K ) g, ) cp( K ) ) hand sde wth respect to, = K P ( K ) > 0. Hence, the left hand sde s strctly ncreasng n. Therefore, hgher data rate should be used when the channel has hgher power gan. Suppose g >g, and the correspondng optmal modulaton and codng result n data rates and respectvely. Hence, U ( ) >U ( ). Besdes, U ( ) = > = U P C + KP ( /K) g P C + KP ( /K) ( ). g Hence, the energy effcency ncrease wth channel gan. Accordng to Theorem, P T ( ) P T ( ) = P C. The dervatve of the left hand sde s P T ( ) > 0. Hence, ncreases wth P C. The proof that the energy effcency decreases wth crcut power s smlar to the proof that energy effcency ncreases wth channel gan. When P C =0, accordng to proof n B, U() s maxmzed when approaches zero,.e. U max = lm 0 P T () = P (0). T = Kr and P T () = KP T ( K ),wherep T (r) s the transmt power on each subchannel, and s monotoncally ncreasng and strctly convex n r. Accordng to Theorem, we have P T ( K )=P C + KP T ( K ), whch s equvalent to r P T (r ) P T (r )= PC K. The left hand sde s ncreasng n r whle the rght hand sde s decreasng n K. Hence, the data rate on each subchannel should decrease wth ncreasng number of subchannels assgned. The proof that the energy effcency ncreases wth the number of subchannels assgned s also smlar to the proof n C and s omtted. The hghest energy effcency s obtaned wth nfnte number of subchannels,.e. U() = lm K P C +P T () = r. Smlar to P T (r) the proof n B, U() s maxmzed when r approaches zero. We have U max = lm r 0 r = P T (r) P (0). T APPENDIX D POOF OF THEOEM 3 Proof: The global convergence s straghtforward from Lemmas. Snce r [0] = αr [0] and r [] r r [], wth nducton, we have r [] r[] = r[0] r[0] (α )r. Hence, ˆr [] = r[] +r[] (r [] (α )r )/ r (α )r and ˆr [] + r + (α )r.thenˆr [] r (α )r +.Let (α )r + ε. We + have log ( (α )r ε ). Theorem 3 follows mmedately. EFEENCES []. G. Gallager, Informaton Theory and elable Communcaton. John Wley & Sons, Inc., 968. [] S. Sampe, S. Komak, and N. Mornaga, Adaptve modulaton/tdma scheme for personal multmeda communcaton systems," n Proc. 994 IEEE Global Telecommun. Conf., Nov. 994, pp [3] M. Ouch, H. J. Lee, S. Komak, and N. Mornaga, Proposal for modulaton level controlled rado system appled to atm networks," n Proc. Fourth European Conf. ado elay Syst., Oct. 993, pp [4] K. Lahr, A. aghunathan, S. Dey, and D. 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10 554 IEEE TANSACTIONS ON COMMUNICATIONS, VOL. 58, NO., FEBUAY 00 [5] J. M. Coff, A Multcarrer Prmer. ANSI TE, 999. [6] C. E. Shannon, Communcaton n the presence of nose," Proc. IE, vol. 37, Jan. 949, pp. 0-. [7] S. Cu, A. Goldsmth, and A. Baha, Energy-effcency of MIMO and cooperatve MIMO technques n sensor networks," n IEEE J. Sel. Areas Commun., vol., no. 6, Aug. 004, pp [8]. G. Gallager, Power lmted channels: codng, multaccess, and spread spectrum," n Proc. Conf. Inf. Sc. Syst., vol., Mar [9] D. Goodman and N. Mandayam, Power control for wreless data," IEEE Wreless Commun., vol. 7, no., pp , Apr [0] N. Feng, S. C. Mau, and N. B. Mandayam, Prcng and power control for jont network-centrc and user-centrc rado resource management," IEEE Trans. Commun., vol. 5, no. 9, pp , Sept [] E. Wolfstetter, Topcs n Mcroeconomcs: Industral Organzaton, Auctons, and Incentves. Cambrdge Unversty Press, 999. [] B. Prabhakar, E. U. Bykoglu, and A. E. Gamal, Energy-effcent transmsson over a wreless lnk va lazy packet schedulng," n Proc. IEEE Infocom 00, vol., 00, pp [3] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge Unversty Press, 004. [4] K. C. Kwel and K. Murty, Convergence of the steepest descent method for mnmzng quasconvex functons," n J. Optmzaton Theory Appl., vol. 89, no., pp. -6, Sept [5] A. J. Goldsmth and S. G. Chua, Varable-rate varable-power MQAM for fadng channels," IEEE Trans. Commun., vol. 45, no. 0, pp. 8-30, Oct [6] ITU- ecommendaton M.5, Gudelnes for evaluaton of rado transmsson technologes for mt-000," 997. Guowang Mao receved the B.S. and M.S. degrees, n 003 and 006, n electronc engneerng from Tsnghua Unversty, Bejng, Chna, and M.S. degree, n 009, n electrcal and computer engneerng from Georga Insttute of Technology, Atlanta, GA, USA. He s currently a Ph.D. canddate at the School of Electrcal and Computer Engneerng, Georga Insttute of Technology. Hs research nterests are n wreless communcatons and networkng, wth a current focus on energy effcent wreless communcatons, optmzaton of dstrbuted random access, and PHY-MAC cross-layer desgn for wreless networks. Nageen Hmayat s a senor research scentst wth Wreless Communcatons and Archtecture Lab at Intel. Her work focuses on algorthm development and system desgn for next generaton wreless systems, ncludng WMAX and 3GPP-LTE famly of standards. Her research nterests are n PHY-MAC cross layer desgn, energy-effcent desgn, MIMO- OFDM technques and n the general area of communcaton theory and statstcal sgnal processng. Pror to jonng Intel, Dr. Hmayat has held postons wth Lucent Technologes and General Instrument Corp. (presently Motorola Corp.). Dr. Hmayat obtaned her B.S.E.E degree from ce Unversty, TX and her Ph.D. degree from the Unversty of Pennsylvana, PA, n 989 and 994 respectvely. Geoffrey Ye L receved hs B.S.E. and M.S.E. degrees n 983 and 986, respectvely, from the Department of Wreless Engneerng, Nanjng Insttute of Technology, Nanjng, Chna, and hs Ph.D. degree n 994 from the Department of Electrcal Engneerng, Auburn Unversty, Alabama. He was a Teachng Assstant and then a Lecturer wth Southeast Unversty, Nanjng, Chna, from 986 to 99, a esearch and Teachng Assstant wth Auburn Unversty, Alabama, from 99 to 994, and a Post-Doctoral esearch Assocate wth the Unversty of Maryland at College Park, Maryland, from 994 to 996. He was wth AT&T Labs - esearch at ed Bank, New Jersey, as a Senor and then a Prncpal Techncal Staff Member from 996 to 000. Snce 000, he has been wth the School of Electrcal and Computer Engneerng at Georga Insttute of Technology as an Assocate and then a Full Professor. He s also holdng the Cheung Kong Scholar ttle at the Unversty of Electronc Scence and Technology of Chna snce March 006. Hs general research nterests nclude statstcal sgnal processng and telecommuncatons, wth emphass on OFDM and MIMO technques, crosslayer optmzaton, and sgnal processng ssues n cogntve rados. In these areas, he has publshed about 00 papers n refereed journals or conferences and fled about 0 patents. He also has two books, enttled, Blnd Equalzaton and Identfcaton (co-authored wth Z. Dng, publshed by Mercel Dekker, Inc. n 000) and OFDM for Wreless Communcatons (co-authored wth G. Stüber, publshed by Sprnger n 006). He s actve n professonal socetes. He once served or s currently servng as an edtor, a member of edtoral board, and a guest edtor for nne techncal journals. He organzed and chared many nternatonal conferences, ncludng techncal program vce-char of the IEEE 003 Internatonal Conference on Communcatons. He has been awarded an IEEE Fellow for hs contrbutons to sgnal processng for wreless communcatons n 005 and selected as a Dstngushed Lecturer from by IEEE Communcatons Socety.