Distributed Resource Allocation and Scheduling in OFDMA Wireless Networks
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1 Southern Illnos Unversty Carbondale OpenSIUC Conference Proceedngs Department of Electrcal and Computer Engneerng Dstrbuted Resource Allocaton and Schedulng n OFDMA Wreless Networks Xangpng Qn xqn@su.edu Randall Berry Northwestern Unversty Follow ths and addtonal works at: Publshed n Qn, X., & Berry, R. (2006). Dstrbuted resource allocaton and schedulng n OFDMA wreless networks. Forteth Aslomar Conference on Sgnals, Systems and Computers, ACSSC '06, do: /ACSSC IEEE. Personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve works for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths work n other works must be obtaned from the IEEE. Ths materal s presented to ensure tmely dssemnaton of scholarly and techncal work. Copyrght and all rghts theren are retaned by authors or by other copyrght holders. All persons copyng ths nformaton are expected to adhere to the terms and constrants nvoked by each author's copyrght. In most cases, these works may not be reposted wthout the explct permsson of the copyrght holder. Recommended Ctaton Qn, Xangpng and Berry, Randall, "Dstrbuted Resource Allocaton and Schedulng n OFDMA Wreless Networks" (2006). Conference Proceedngs. Paper Ths Artcle s brought to you for free and open access by the Department of Electrcal and Computer Engneerng at OpenSIUC. It has been accepted for ncluson n Conference Proceedngs by an authorzed admnstrator of OpenSIUC. For more nformaton, please contact opensuc@lb.su.edu.
2 Dstrbuted Resource Allocaton and Schedulng n OFDMA Wreless Networks. Xangpng Qn Samsung Informaton Systems Amerca San Jose, CA x.qn@samsung.com Randall Berry Northwestern Unversty Evanston, IL rberry@ece.northwestern.edu Abstract In ths paper we develop dstrbuted resource allocaton and schedulng algorthms for the uplnk of an orthogonal frequency dvson multple access (OFDMA) wreless network. We consder a tme-slotted model, where n each tme-slot the users are assgned to subchannels consstng of groups of OFDM tones. Each user can also allocate ts transmsson power among the subchannels t s assgned. We consder dstrbuted algorthms for accomplshng ths, where each user s actons depend only on knowledge of ther own channel gans. Assumng a collson model for each subchannel, we characterze an optmal polcy whch maxmzes the system throughput and also gve a smpler sub-optmal polcy. We study the scalng behavor of these polces n several asymptotc regmes for a broad class of fadng dstrbutons. I. INTRODUCTION It s well establshed that dynamcally allocatng transmsson resources can mprove the performance of wreless networks. In ths paper, we consder these approaches for the uplnk n a wreless access network whch uses orthogonal frequency dvson multple access (OFDMA), such as n the IEEE (WMAX) standard. In OFDMA networks the prmary resources are the assgnment of tones or subcarrers to users and the allocaton of a user s power across her assgned tones. Such resource allocaton problems have been wdely studed, e.g. see [1] [4]. Most of ths pror work focuses on the case n whch resource allocaton decsons are made by a centralzed controller wth knowledge of every user s channel state. Because of the requred overhead and delays nvolved, t may not be feasble to acqure ths nformaton n a fast-fadng envronment or a system wth a large number of users and/or subcarrers. Here, we nstead consder approaches where each transmtter allocates ts transmsson rate and power based only on knowledge of ts own channel condtons. Ths can be obtaned, for example, va a sngle plot sgnal broadcast by the recever n a tme-dvson duplex system [5]. Ths requres much less overhead, but snce each user has ncomplete nformaton, a dstrbuted approach for resource allocaton s requred. In pror work [5], [6], we have consdered a dstrbuted schedulng approach based on the Aloha protocol for the Ths work was supported n part by the Northwestern-Motorola Center for Seamless Communcatons and by NSF CAREER award CCR Ths work was performed whle X. Qn was wth the Department of EECS, Northwestern Unversty. case where all users communcate over a sngle flat-fadng channel. In ths approach each user randomly transmts wth a probablty based on ts own channel gan. It s shown that as the number of users ncreases, the throughput of such a system scales at the same rate as that obtaned by an optmal centralzed controller. In [7], we extended ths approach to an OFDMA-type of system, where each user can transmt over multple subchannels, and can allocate transmsson power across these subchannels. In [7], the asymptotc analyss was restrcted to the case where each subchannel had..d. Raylegh fadng. In ths paper, we extend ths analyss to a larger class of fadng dstrbutons, whch ncludes Raylegh, Rcean and Nakagam fadng. II. MODEL DESCRIPTION We consder a model of n users communcatng to a sngle recever. There are k avalable subchannels. Each subchannel may represent a sngle OFDM tone, or more lkely a group of dsjont tones bundled together. 1 Each subchannel between each user and the recever s modeled as a tme-slotted, blockfadng channel wth frequency-flat fadng and bandwdth W c. Ths s reasonable when all the tones n a subchannel le wthn a sngle coherence band; when ths s not the case, then ths can be vewed as an approxmaton n whch the channel gan represents the average gan for the subchannel. At each tme t, the receved sgnal on the jth subchannel s gven by n y j (t) = H j (t)x j (t)+z j (t), (1) where x j (t) and H j (t) are the transmtted sgnal and channel gan for the th user on subchannel j, and z j (t) s addtve whte Gaussan nose wth power spectral densty N0 2. To smplfy notaton we assume that N 0 W c =1. The channel gans are assumed to be fxed durng each tme-slot and to randomly vary between tme-slots,.e. H j (t) =H j for all t [mt, (m+1)t], where T s the length of a tme-slot. Here, {H j },..,n,,..,k are assumed to be ndependent and dentcally dstrbuted (..d.) across both the users and subchannels 1 For example n , subchannels are formed by groupng a set of nterleaved tones (the default mode) or by groupng adjacent tones (n the optonal Band AMC mode) /06/$
3 wth a contnuous probablty densty f H (h) on [0, ). 2 We assume that E(H,k ) < and that f H (h) > 0, for all h>0 and s dfferentable. It follows that the correspondng dstrbuton functon F H (h) s strctly ncreasng and twce dfferentable. Let FH (h) =1 F H (h) denote the channel gan s complmentary dstrbuton functon. For example, f each subchannel experences Raylegh fadng, then H wll be exponentally dstrbuted, and so F H (h) =e h/h0, where h 0 = E(H,k ). We focus on the case where at the start of each slot, each user has perfect knowledge of H 1,..., H k, but no knowledge of the channel gans for any other users. For convenence, we drop the user subscrpt and let H =(H 1,..., H k ) denote the vector of channel gans for an arbtrary user. Let P(h) = (P 1 (h),p 2 (h),..., P k (h)) be a user s power allocaton, where P j (h) ndcates the power allocated to subchannel j gven that H = h. 3 Ths power allocaton must satsfy a total power constrant of ˇP across all subchannels n each tmeslot,.e., j P j(h) ˇP, for all h. No cooperaton exsts among users. In partcular, all users are requred to employ the same power allocaton and transmsson scheme;.e., they can not cooperate n selectng these allocatons. Durng each tme-slot, we assume that at most one user can successfully transmt on each subchannel. If more than one user transmts on a gven subchannel, a collson occurs and no packets are receved. However, a packet sent over another subchannel wthout a collson wll stll be receved,.e., the nformaton sent over each subchannel s ndependently encoded. Gven that only one user transmts on subchannel j, let R(γ j ) ndcate the rate at whch the user can relably transmt as a functon of the receved power γ j = h j P j (h). We assume that R(γ) := log(1 + γ), whch s proportonal to the Shannon capacty of the subchannel durng a gven tmeslot. We assume that there s no codng done over successve tme-slots. Also, we do not consder any multuser recepton or power capture effects when multple users transmt on a subchannel. III. OPTIMAL DISTRIBUTED POWER ALLOCATION Next we turn to the power allocaton P(h) used by each user durng each tme-slot. To begn, consder the case where there s only n =1user who must allocate ts power over the k avalable subchannels. In ths case, for each channel realzaton h, the power allocaton that maxmzes a user s throughput s the well-known water-fllng allocaton, P j (h) =(λ 1 h j ) +, where λ s chosen so that k P j(h) = ˇP. When there are multple users, f more than one user transmts on a subchannel, a collson results and no data s receved. Followng [7], we consder an Aloha-based approach, where each user transmts on each subchannel wth a certan probablty p. Snce each subchannel s..d., t s reasonable to requre that each user transmts wth the same probablty 2 In an OFDM system dfferent sub-carrers wll typcally experence correlated fadng. However, f each subchannel s a large enough group of sub-carrers, then ths ndependence assumpton s reasonable. 3 If a user does not transmt on channel j, then P j (h) =0. p n each slot and on each subchannel. The probablty of some user successfully transmttng on one subchannel s then np(1 p) n 1. Gven ths probablty, for each subchannel j, each user chooses a subset H j of the possble realzatons of H wth Pr(H H j )=p. The user then only transmts on subchannel j when H H j. To maxmze the total throughput, each user wll choose channel states n each set H j that can acheve hgher transmsson rates. However, the transmsson rate that can be acheved also reles on the specfc power allocaton, e.g. f a state h s n both H j and H l, the user must allocate power across both subchannels, whle f h s n only one set, the user can use all the avalable power on the correspondng subchannel. For a gven power allocaton, P j (h), the expected transmsson rate on subchannel j, condtoned on a user successfully transmttng on that subchannel s gven by ( E H R(Hj P j (H)) ) H Hj ( = E H R(Hj P j (H)) Pj (H) > 0 ), where we have used that the channel gans are ndependent across users. We now specfy the followng dstrbuted optmal throughput problem: ( np(1 p)n 1 E H R(Hj P j (H)) P j (H) > 0 ) max P(H),p s.t. P j (h) ˇP, h Pr{P j (H) > 0} = p, j =1,..., k. The objectve n (2) s the average sum throughput for all n users over all k subchannels. Ths s optmzed over the transmsson probablty p and the power allocaton (P 1 (H),P 2 (H),..., P k (H)), whch s used by each user. The second constrant ensures that the sets H j all have probablty p. When ths constrant s met, t follows that ( pe H R(Hj P j (H)) P j (H) > 0 ) = E H (R(H j P j (H)). Hence, the objectve n (2) can also be wrtten as n(1 p) n 1 E H (R(H j P j (H)). (3) For a gven channel realzaton h, let (h (1),h (2),..., h (k) ) denote the ordered channel gans from the largest to the smallest, wth any tes broken arbtrarly. It can be shown that the soluton to (2) s symmetrc and so t wll just depend on ths ordered sequence n each tme-slot. Gven ths ordered sequence, for j l k, let R(j) l (h) denote the rate achevable over the jth best channel when the transmtter uses the optmal (water-fllng) power allocaton over only the l best channels. In other words, R(j) l (h) = log(1 + P (j)(h)h (j) ), where P (j) (h) = (λ 1 h (j) ) + and λ s chosen such that l P (j)(h) = ˇP. Lemma 1: As l ncreases, l Rl () (h) l 1 Rl 1 () (h) s non-ncreasng. (2) 1943
4 Gven a threshold rate R th > 0 for each channel realzaton h, we ntroduce the followng problem: max s.t. l=1,...,k l l l 1 R() l (h) R l 1 () (h) R th If ths problem has no feasble soluton, we defne the soluton to be l =0. When k =1, the constrant n (4) s R(1) 1 (h) R th,.e., the rate when only transmttng on the best channel should be greater than R th.fork =2, the constrant n (4) becomes R(1) 2 (h)+r2 (2) (h) R1 (1) (h) R th, whch means that the ncrease n the total rate from usng the best two channels versus only usng the best channel should be greater than R th. In general, the objectve of (4) s to fnd the maxmal number of channels l, such that the gan n the sum rate from transmttng on the l best channels nstead of only the l 1 best channels s at least R th. From Lemma 1 t follows that f l solves (4), then any l<l wll also be feasble. For a gven R th, let P R th (h) be the power allocaton that corresponds to solvng (4) for each channel realzaton h;.e. ths wll be a water-fllng allocaton over the l best channels, where l s the soluton to (4) for each gven realzaton (note l may change wth each realzaton). The followng proposton relates ths to the soluton of (2). Proposton 1: There exsts a constant R th > 0 such that P R th (h) s also the optmal soluton to (2). Ths proposton specfes the form of the optmal power allocaton; the correspondng transmsson probablty s gven by p =Pr(P R th (H) > 0). It follows from ths proposton that the optmal soluton to (2) can be found by solvng (4) for a gven R th, and then teratvely searchng for the optmal R th. An algorthm for solvng (4) for a gven R th and channel realzaton h s gven n [7]. Ths algorthm uses the property n Lemma 1, to converge to the optmal soluton to (4) n at most k teratons. The optmal value of R th must stll be found va a numerc search; however, we note that ths search s now only a one-dmensonal search, nstead of a k-dmensonal search over the possble power allocatons. For a gven n and k, the optmal power allocaton could be determned offlne usng ths procedure. For a large number of channels k ths wll result n a large computatonal cost. Next, we ntroduce a smpler sub-optmal algorthm and analyze ts performance. IV. SUB-OPTIMAL POWER ALLOCATION AND ASYMPTOTIC ANALYSIS We consder a smplfed dstrbuted scheme, where nstead of fndng a threshold rate R th and solvng (4), we set a threshold h th on the channel gan. Each user then transmts on the kth subchannel when ts gan s greater than h th, resultng n the transmsson probablty p = F H (h th ).Ifa user has more than one subchannel whose gan s hgher than the threshold, then the total power ˇP wll be allocated equally to each of these subchannels. 4 Gven that a user transmts on 4 Other smlar equal power allocaton approaches for mult-carrer systems have been studed, see e.g. [8]. (4) subchannels, we assume t transmts at a constant rate of 1 ˇP R (p) :=R( F H (p) ) on each subchannel. Ths s a lower bound on the achevable rate and smplfes our analyss. The total throughput usng ths scheme s a functon of k, n and p. For =1,..., k, let q k,p () be the probablty one user 1 has subchannels above the threshold h th = F H (p),.e., ( ) k q k,p () = (p) (1 p) k. Among these subchannels, for j =1,...,, let ω p, (j) be the probablty a user transmts successfully on exactly j subchannels,.e. the probablty there s no collson on exactly j subchannels, gven that are above the threshold. Ths s gven by ( ) [(1 ωp,(j) n = p) n 1 ] j [ 1 (1 p) n 1 ] j. j The average sum throughput of ths system s then gven by s(k, n, p) =n q k,p () ωp,(j)jr n (p). Note that ωp, n (j) s a bnomal probablty mass functon (p.m.f.) and so ωn p, (j)j =(1 p)n 1. Therefore, s(k, n, p) =n(1 p) n 1 ( ) k (p) (1 p) k R (p). (5) We consder how the sum throughput of ths scheme and the optmal dstrbuted scheme scales n three asymptotc regmes. We defne two sequences f(m) and g(m) to be asymptotcally f(m) equvalent, denoted by f(m) g(m), flm m g(m) = c. In the specal case where c =1, we say that they are strongly asymptotcally equvalent and denote ths by f(m) g(m). Ths mples that both sequences asymptotcally grow at the same rate and have the same frst order constant. For our analyss, we make an addtonal assumpton on the tal of the fadng dstrbuton. Specfcally, we assume that as h, f H (h) f H(h), (6) where f H (h) = d dh f H(h). Ths s satsfed by any fadng dstrbuton that has an exponental tal, whch s the case for most common fadng models such as Raylegh, Rcean and Nakagam fadng. Lemma 2: For any contnuous, dfferentable fadng densty that satsfes (6), then the followng condtons hold: f H FH(h) (a.) FH (h) f H (h), (b.) lm ] h hf H(h) =0. lm h d dh [ FH(h) f H(h) = 0, and (c.) These condtons follow drectly from evaluatng the lmts usng L Hosptal s rule. We also compare the dstrbuted approaches to an optmal centralzed system that maxmzes the throughput n every slot. 1944
5 Ths s gven by: 5 max n {P j,c j} s.t. R(P j c j h j ) P j c j = ˇP,, n c j 1, j, c nk {0, 1},, j. Here, the nteger varables, c j, ndcate when user s assgned to subchannel j; the second constrant ensures that at most one user s assgned to each subchannel. Let s ct (k, n) be the average sum throughput obtaned by the optmal centralzed schedulng polcy. Denote the throughput of the optmal dstrbuted polcy by s (k, n) and the optmal throughput of the threshold-based algorthm by s(k, n, p ), where p s the transmsson probablty that optmzes s(k, n, p). For all n and k, wehave, (7) s ( k, n, 1 n) s(k, n, p ) s (k, n) s ct (k, n), (8) where the frst term s the throughput wth a transmsson probablty of 1/n. Frst, we consder the case where k s fxed and n ncreases. Proposton 2: Gven any fnte k, asn, s(k, n, 1 n ), s(k, n, p ), s (k, n) and 1 e s ct(k, n) are all strongly asymptotcally equvalent to k e log ( 1 1+ ˇP F H ( 1 n )). In other words, asymptotcally there s no dfference n the frst-order performance compared to the optmal dstrbuted approach when usng the smplfed scheme or from choosng p = 1 n nstead of the optmal p. The throughput for each dstrbuted approach asymptotcally ncreases lke k 1 e log(1+ ˇP F H ( 1 n )), as does 1 e tmes the throughput wth the optmal centralzed scheduler. In other words, the dstrbuted approaches all grow at the same rate as the centralzed approach and asymptotcally the rato of ther throughputs approach 1 e, the contenton loss n a standard slotted Aloha system. As an example, for the case of..d. Raylegh fadng on each channel the throughput n each case wll ncrease at rate O(log(log(n)). The second regme we consder s when n s fxed and k ncreases. Proposton 3: Gven any fnte n, ask, s(k, n, p ), s (k, n), s ct (k, n) are all strongly asymptotcally equvalent 1 to n ˇP F H ( 1 k ). Agan the threshold based approach s strongly asymptotcally equvalent to the optmal dstrbuted approach. In ths case, t s also asymptotcally equvalent to the optmal centralzed system;.e. there s no loss of 1 e. Intutvely, ths s because as the number of channels ncreases, the probablty of collson becomes neglgble. In ths case, for a Raylegh fadng channel each of these terms grows lke O(log(k)) as k, wth a frst order constant that s lnear n n. 5 Ths s smlar to a problem studed for centralzed OFDM systems n [3]. The last regme we consder s where both k and n ncrease wth fxed rato k n = β. k Proposton 4: If n = β, as n, s(βn,n, 1 n ), s(βn,n,p ), s (βn,n) and 1 e s ct(βn,n) are all strongly asymptotcally equvalent to βne 1 log ( 1 1+ ˇP F H ( 1 n )). As n Proposton 2, once agan compared to the centralzed scheme there s an asymptotc penalty of 1/e due to the contenton, and a transmsson probablty of p = 1 n s asymptotcally optmal for the dstrbuted system. For Raylegh fadng channels the throughput now grows lke O(n log(log(n))), as n, wth a frst order constant that s lnear n β. V. NUMERICAL EXAMPLES We next gve some numercal examples to llustrate the performance of the optmal and smplfed dstrbuted algorthms wth a fnte number of channels and users. All the results n ths secton are for an..d. Raylegh fadng model, wth E(H j ) = 1, and a total power constrant of ˇP = 1. The performance s averaged over multple channel realzatons. Fgure 1 shows the average throughput acheved by the optmal dstrbuted power allocaton scheme from Secton III compared to the smplfed power allocaton scheme n Secton IV. The throughput of both approaches s shown as a functon of the number of users for a system wth k =10channels. As the number of users ncreases, both throughputs ncrease and the dfference between the two curves decreases. Fgure 2 shows upper and lower bounds on the rato of the average throughput of the optmal dstrbuted scheme s (k, n) to the centralzed scheme s ct (k, n) defned n (7) as a functon of the number of users, for k = 5 and 10 channels. Calculatng s ct (k, n) requres solvng the optmzaton problem n (7) for every channel realzaton, whch s complcated due to the nteger constrants. Instead we compare s (k, n) to upper and lower bounds on s ct (k, n). We upper bound s ct (k, n) by relaxng the total power constrant on the channels, k P nkc nk = ˇP. Instead, we allow each user to transmt wth P nk = ˇP over each channel. The maxmum throughput s acheved for ths relaxed system by lettng the best user on each channel transmt at each tme. To lower bound s ct (k, n), we stll choose the best user to transmt on each subchannel, but f one user s chosen to transmt on more than one subchannel, ts power s dvded equally across these channels. The resultng throughput s then a lower bound on s ct (k, n). Fgure 2 shows that as the number of users ncreases, the two bounds approach each other. It can be seen that the rato of the throughputs of the dstrbuted to the optmal scheme s decreasng as the number of users ncreases and s larger than the lmtng value of 1/e (see Proposton 2) for all fnte n. As the number of the channels, k, ncreases, the throughput rato also ncreases for a fxed n. Ths s due to the ncreased frequency dversty wth more channels. Fgure 3 shows upper and lower bounds on the rato of the throughput of the optmal dstrbuted scheme to that of the optmal centralzed approach as the number of channels ncreases, for a system wth n =5and 10 users. In ths case, we upper bound s ct (k, n) by the nformaton theoretc capacty 1945
6 Average Throughput (bps) Smplfed Optmal Number of users Throughput Rato Lower bound n= Lower bound n= Upper bound n=10 Upper Bound n= Number of Channels Fg. 1. Average throughput (bps) per channel of the optmal dstrbuted scheme and the smplfed dstrbuted scheme as a functon of the number of users for k =10channels. Fg. 3. Lower and upper bounds on the rato of the average throughputs of the optmal dstrbuted scheme to the optmal centralzed scheme versus the number of channels, for n =5and 10 users. Throughput Rato Lower bound Upper bound k= Number of Users Fg. 2. Lower and upper bounds on the rato of average throughputs of the optmal dstrbuted scheme to the optmal centralzed scheme versus the number of users, for k =5and 10 channels. of ths mult-access system. In other words, jont decodng s used when multple users transmt on the same channel. We lower bound s ct (k, n) by only allowng the user who has the best channel to transmt on a channel. Fgure 3 shows that as the number of channels ncreases, the two bounds quckly converge. The throughput rato ncreases as the number of channels ncreases. From Proposton 3, as k ncreases, these bounds should approach 1. In ths asymptotc regme, the convergence appears to be much slower than n Fgure 2. VI. SUMMARY We have presented dstrbuted algorthms for resource allocaton n an OFDMA wreless network, where each user only has knowledge of ts own channel gans. Usng a contenton model, an optmal dstrbuted algorthm s characterzed. A smplfed dstrbuted approach s also gven. In three dfferent k=5 1/e asymptotc regmes, the smplfed algorthm s shown to be asymptotcally equvalent to the optmal dstrbuted algorthm. Both algorthms are also shown to scale at the same rate as the optmal centralzed scheduler. These results suggest that t s possble to develop near optmal approaches for schedulng and power allocaton wthout requrng a centralzed controller wth complete channel knowledge. There are several mportant ssues that we have not addressed here. For example, we have not consdered asymmetrc models, where the fadng s not dentcally dstrbuted across the channels or the users, or models where the fadng s correlated across the channels. We also assumed that each user knows the fadng dstrbuton; n practce, an adaptve approach would be requred to estmate ths dstrbuton. REFERENCES [1] W. Yu, W. Rhee, S. Boyd and J. Coff, Iteratve Water-fllng for Gaussan Vector Multple Access Channels, n Proceedngs of IEEE Internatonal Symposum on Informaton Theory, [2] W. Yu and J. Coff, FDMA Capacty of Gaussan Multple Access Channels wth ISI, IEEE Trans. on Communcatons, vol. 50, pp , [3] C. Y. Wong, R. S. Cheng, K. B. Letaef, and R. D. Murch, Multuser OFDM wth adaptve subcarrer, bt, and power allocaton, IEEE Journal on Selected Areas n Communcatons, Vol. 17, pp , [4] M. Ergen, S. Coler, and P. Varaya, QoS Aware Adaptve Resource Allocaton Technques for Far Schedulng n OFDMA Based Broadband Wreless Access Systems, IEEE Trans. on Broadcastng, Vol. 49 pp , [5] X. Qn and R. Berry, Explotng Multuser Dversty for Medum Access Control n Wreless Networks, n Proceedngs of 2003 IEEE INFOCOM, pp , March [6] X. Qn and R. Berry, Dstrbuted Approaches for Explotng Multuser Dversty n Wreless Networks, IEEE Transactons on Informaton Theory, Vol. 52, pp , [7] X. Qn and R. Berry, Dstrbuted Power Allocaton and Schedulng for Parallel Channel Wreless Networks, n Proc. of 3rd Intl. Symposum on Modelng and Optmzaton n Moble, Ad Hoc, and Wreless Networks (WOpt) Trentno, Italy, Aprl [8] L. Hoo, B. Halder, J. Tellado and J. Coff, Multuser Transmt Optmzaton for Multcarrer Broadcast Channels: Asymptotc FDMA Capacty Regon and Algorthms, IEEE Transacton on Communcatons, Vol. 52, pp , June
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