Unversty of Wollongong Research Onlne Faculty of Informatcs - Papers (Archve) Faculty of Engneerng and Informaton Scences 2006 Subcarrer allocaton for OFDMA wreless channels usng lagrangan relaxaton methods Gengfa Fang Chnese Academy of Scences Y Sun Chnese Academy of Scences Jhua Zhou Chnese Academy of Scences Jngln Sh Chnese Academy of Scences Zhongcheng L Chnese Academy of Scences See next page for addtonal authors Publcaton Detals Fang, G., Sun, Y., Zhou, J., Sh, J., L, Z. & Dutkewcz, E. (2006). Subcarrer allocaton for OFDMA wreless channels usng lagrangan relaxaton methods. IEEE Global Telecommuncatons Conference (pp. 1-5). Pscataway, USA: IEEE. Research Onlne s the open access nsttutonal repostory for the Unversty of Wollongong. For further nformaton contact the UOW Lbrary: research-pubs@uow.edu.au
Subcarrer allocaton for OFDMA wreless channels usng lagrangan relaxaton methods Abstract In ths paper, we propose a practcally effcent Subcarrer Allocaton scheme based on Lagrangan relaxaton to solve the problem of subcarrer allocaton n OFDMA wreless channels. The problem of subcarrer allocaton s formulated nto an Integer Programmng (IP) problem, whch s relaxed by replacng complcatng constrants wth Lagrange multplers usng Lagrangan Relaxaton. A subgradent method s used to optmze the Lagrangan dual functon and a heurstc s desgned to obtan the feasble soluton. Lagrangan Relaxaton Subcarrer Allocaton (LRSA) s proven to be of polynomal complexty and t provdes bounds on the value of channel effcency. Numercal results show that compared wth other algorthms proposed n the lterature, LRSA can result n a sgnfcant mprovement n channel effcency, whle at the same tme guaranteeng mnmum data rates of users. Dscplnes Physcal Scences and Mathematcs Publcaton Detals Fang, G., Sun, Y., Zhou, J., Sh, J., L, Z. & Dutkewcz, E. (2006). Subcarrer allocaton for OFDMA wreless channels usng lagrangan relaxaton methods. IEEE Global Telecommuncatons Conference (pp. 1-5). Pscataway, USA: IEEE. Authors Gengfa Fang, Y Sun, Jhua Zhou, Jngln Sh, Zhongcheng L, and E. Dutkewcz Ths conference paper s avalable at Research Onlne: http://ro.uow.edu.au/nfopapers/2995
Subcarrer Allocaton for OFDMA Wreless Channels Usng Lagrangan Relaxaton Methods Gengfa Fang, Y Sun, Jhua Zhou, Jngln Sh, Zhongcheng L Insttute of Computng Technology Chnese Academy of Scences, Bejng, Chna {gengfa.fang, suny, jhzhou, sjl, zcl}@ct.ac.cn Eryk Dutkewcz Wreless Technologes Laboratory Unversty of Wollongong, Wollongong, Australa eryk@uow.edu.au Abstract In ths paper, we propose a practcally effcent Subcarrer Allocaton scheme based on Lagrangan relaxaton to solve the problem of subcarrer allocaton n OFDMA wreless channels. The problem of subcarrer allocaton s formulated nto an Integer Programmng (IP) problem, whch s relaxed by replacng complcatng constrants wth Lagrange multplers usng Lagrangan Relaxaton. A subgradent method s used to optmze the Lagrangan dual functon and a heurstc s desgned to obtan the feasble soluton. Lagrangan Relaxaton Subcarrer Allocaton (LRSA) s proven to be of polynomal complexty and t provdes bounds on the value of channel effcency. Numercal results show that compared wth other algorthms proposed n the lterature, LRSA can result n a sgnfcant mprovement n channel effcency, whle at the same tme guaranteeng mnmum data rates of users. I. INTRODUCTION Unlke wrelne channels, channels n broadband wreless access (BWA) networks are prone to frequency selectve fadng. Orthogonal Frequency Dvson Multple Access (OFDMA) s a modulaton and multple access method used n IEEE 802.16 [1] and IEEE 802.11a and s consdered for usenemergng4 th generaton wreless networks because of ts hgh mmunty to nter-symbol nterference and frequency selectve fadng. Unlke OFDM systems, where only a sngle user s allowed to use all of the subcarrers at any gven tme, OFDMA allows multple users to share the subcarrers, whch brngs new challenges to the resource allocaton problem n OFDMA wreless channels. OFDMA s based on the Orthogonal Frequency Dvson Multplexng (OFDM) technque where the total avalable bandwdth s dvded nto several narrow subcarrers. OFDMA may result n hgher channel effcency by explotng the frequency dversty among subcarrers under the control of a properly desgned subcarrer allocaton algorthm. The man goal of subcarrer allocaton n OFDMA wreless channels s, on one hand, to mprove the average channel effcency by explotng both the tme dversty n the tme doman and frequency dversty n the frequency doman of the wreless channel. On the other hand, the subcarrer allocaton algorthm should guarantee the prescrbed QoS requrements of users, such as the mnmum data rate consdered n ths paper. The problem of subcarer allocaton n OFDMA systems has been recently studed qute extensvely. In [2], subcarrers are allocated to users based on the current channel condton and current buffer state as well as the measured rato of the arrval rate to throughput for each user. In [3], a delay utlty functon based subcarrer allocaton scheme s presented for real tme servces, where the man goal of the proposed scheme s to maxmze the aggregate utlty. However the above schemes do not take any QoS requrements nto account, thus they provde no guarantee of the mnmum servce qualty for users. In reference [4], Wong et. al. present a new method to solve the subcarrer allocaton problem n OFDMA system by proportonally dstrbutng the subcarrers among users based on ther QoS requrements. Wong s algorthm s conservatve n the sense of channel effcency mprovement although t can guarantee a proportonal farness among users. Other works on ths topc.e. [5] [7], try to combne the subcarrer allocaton wth the power allocaton problem, and offer no better soluton to the subcarrer allocaton problem. In ths paper, we focus on subcarrer allocaton n wreless OFDMA channel where users have QoS requrements n terms of the mnmum data rates. The subcarrer allocaton process s formulated nto an Integer Programmng (IP) problem wth constrants. We solve the problem based on the Lagrangan relaxaton theory and propose an effcent new scheme called Lagrangan Relaxaton Subcarrer Allocaton: LRSA. The propertes and performance of LRSA are verfed through smulatons. The rest of the paper s organzed as follows. In Secton II, the system model of subcarrer allocaton n OFDMA wreless channels s ntroduced and the subcarrer allocaton problem s formulated. In Secton III, we analyze the subcarrer allocaton problem and propose our soluton usng Lagrangan relaxaton methods. Smulaton results are gven n Secton IV, whch s followed by our conclusons n Secton V. II. SYSTEM MODEL In ths paper, we consder the downlnk OFDMA cell system, where there s only one base staton (BS) and multple users. The wreless channel between the BS and users conssts of K subcarrers, whch are shared by M users under the control of the subcarrer allocaton algorthm n BS. The users measure the current Channel State Informaton (CSI) based on the receved sgnals and then send SCI back to BS n a predefned feedback channel. Durng each subcarrer allocaton cycle, the scheduler n BS allocates the subcarrers to users Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE GLOBECOM 2006 proceedngs.
based on the SCI and QoS requrements of users. Based on the result of subcarrer allocaton, bts of users data are then modulated nto the subcarrers allocated to them. All the subcarrers belongng to dfferent users are then combned nto a sngle symbol and broadcasted to users. Next, we lst some notatons used n ths paper and then formulate the subcarrer allocaton problem. Index of users n the cell. (1,2,...,M). j Index of subcarrers for the downlnk channel. j (1,2,...,K). c,j The achevable data rate of user at subcarrer j n the current tme slot for a gven BER and transmsson power. x,j Indcaton of subcarrer j allocaton to user. x,j equals to 1 f subcarrer j s allocated to user, R mn r r + ˆr mn otherwse, x,j s equal to 0. Mnmum data rate n bts per second that user should receve n order to guarantee ts servce qualty. Ths s the only QoS parameter consdered n ths paper. The data rate that user has receved before the current subcarrer allocaton cycle. The data rate of user after the current subcarrer allocaton cycle. The mnmum data rate that user should get n the current subcarrer allocaton cycle n order to meet ts QoS requrement. The total bandwdth of the wreless channel s B Hz and f we use f Hz to stand for the bandwdth of each subcarrer, we have f = B/K. The achevable data rate of user at subcarrer j,.e., c,j, can be calculated as follows: c,j = f log 2 (1 + βρ,j ) (1) where ρ,j s the sgnal to nose rate (SNR) ofuser on subcarrer j n current tme slot and β = 1.5ln(5 BER) [2][3]. In ths paper, we assume that power s equally dstrbuted over all the subcarrers so that the maxmum feasble data transmsson rate s only decded by the current channel qualty n SNR and BER. We also assume that the scheduler n the base staton knows exactly the channel state nformaton of the subcarrers at the begnnng of each subcarrer allocaton cycle,.e., c,j, s avalable at the begnnng of each schedulng cycle. Based on the defntons above, we defne the subcarrer allocaton problem n each allocaton cycle as follows: There are K subcarrers to be allocated to M users. Each sucarrer j must be allocated to one and only one user. Each user can occupy multple or no subcarrers durng each allocaton cycle. A data rate of c,j n bts per second wll be acheved when subcarrer j s allocated to user. Durng each allocaton cycle, a data rate of ˆr mn n bts per second must be assgned to user as a result of subcarrer allocaton. We have ˆr mn 0. The man goal of subcarrer allocaton n each cycle s to maxmze the overall data rate (or channel effcency) as a result of subcarrers allocatons subject the constrants above. The subcarrer allocaton problem can be then formulated as follows: (IP) s.t. f(x) =max =1 j=1 K c,j x,j (2a) x,j =1; j =1, 2,..., K (2b) =1 K j=1 c,j x,j ˆr mn ; =1, 2,..., M (2c) x,j (0, 1); =1, 2,..., M; j =1, 2,..., K. (2d) Objectve functon f(x) s the total data rate as a result of the current subcarrer allocaton cycle. Constrants (2b) ensure that each subcarrer can only be allocated to one user. Constrants (2c) are the mnmum data rate constrants of users n the current schedulng cycle. III. LRSA SUBCARRIER ALLOCATION In ths secton, we descrbe n detal how to solve the above subcarrer allocaton problem usng Lagrangan relaxaton. A servce trackng and mnmum data rate requrement estmaton method s also proposed. Followng that we present our LRSA algorthm. A. Lagrangan Relaxaton based Subcarrer Allocaton The goal of subcarrer allocaton s to maxmze the total channel effcency whle at the same tme guaranteeng mnmum data rates of users as defned n the IP problem n the prevous secton. Clearly the IP problem s a 0-1 nteger optmzaton problem. Unfortunately, as proven n many references [8], the IP problem s NP-hard. However, by nspecton, we fnd that f we remove the constrants of (2c) n the IP problem, then we get a smplfed IP problem defned as follows: K (IP ) f(x) =max c,j x,j (3a) =1 j=1 s.t. constrants (2b) and (2d). (3b) The soluton of the IP problem above becomes feasble n polynomal tme, where the optmal soluton s to choose a user wth maxmum c,j for each of the subcarrers n current subcarrer j,.e., = arg max c,j ; j =(1, 2,..., K). Constrants of (2c) are called complcatng constrants, whch make the IP problem polynomal tme unfeasble. Based on the above, t s clear that f we can fnd a method whch can remove constrants of (2c) n a specfc way and at the same tme keep the lnear property of the IP problem then the IP problem becomes polynomal tme feasble. Ths can be acheved usng Lagrange relaxaton. Next, we brefly explan the man dea n Lagrangan relaxaton methodology and construct the Lagrangan dual problem. Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE GLOBECOM 2006 proceedngs.
Lagrangan relaxaton s a mathematcal programmng technque for solvng constraned optmzaton problems. The man dea of Lagrange relaxaton s to create a relaxed problem by replacng complcatng constrants wth Lagrange multplers. The relaxed problem then can be decomposed nto subproblems, whch are much easer to solve compared to the orgnal problem and solutons can be effcently obtaned by dynamc programmng. The multplers are then adjusted teratvely based on the degree of constrant volaton whle at the same tme the subproblems are resolved based on the new multplers. In mathematcal terms, a dual functon s maxmzed or mnmzed when the multplers are updated. The value of the dual functon s the lower or upper bound of the orgnal problem. However there are several dffcultes when applyng ths method for solvng dscrete varable problems such as the subcarrer allocaton n the OFDMA channel as defned above. Frstly, the dual functon s not dfferentable everywhere, so specalzed methods for optmzng ths nondfferentable dual functon must be employed, such as the subgradent method used n ths paper. Secondly, even f the optmum of the dual functon s obtaned, the correspondng result at that pont may not be feasble, so adjustments may be needed to make sure that all the constrants are met. Accordng to the defnton of Lagrange relaxaton, we relax the complcatng constrant (2c) n IP usng Lagrange multpler λ ( = 1, 2,..., M) and then get the followng Lagrange Relaxaton problem LR: f LR (λ) =max K c,j (1 + λ )x,j λ ˆr mn =1 j=1 =1 (4a) s.t. constrants (2b) and (2d). (4b) Accordng to constrants (2c), we have: K c,j x,j ˆr mn 0; =(1, 2,..., M). j=1 So we have: λ 0, f LR (λ) f IP (x). (5) Expresson (5) ndcates that LR s an upper bound on the optmal value of the orgnal IP problem and λ >0 can be used to produce the upper bound. In order to get the optmal lowest upper bound, our goal s to make the upper bound as close to the optmal value of the orgnal IP problem as possble. We defne problem LD as the optmal upper bound of IP and we have: (LD) = mn f LD = mn λ 0 f LR(λ) max λ 0 x M (( K c,j (1 + λ )x,j =1 j=1 =1 λ ˆr mn ). Problem LD s also called the Lagrange dual problem of IP, whch s concave and pecewse lnear. Several steps wll be presented n order to get the near optmal soluton. In the next (6) subsectons, we provde the solutons to the subproblems, to the Lagrange dual problem LD and to obtanng a feasble soluton. 1) Solvng Subproblems : Gven Lagrange multpler λ, the Lagrangan dual problem LD leads to the decomposed subproblem for each subcarrer j as follows: (SP) max ( c,j (1 + λ )x,j ) (7) =1 s.t. constrants (2b) and (2d). where the Lagrange multplers can be nterpreted as the gan that user uses subcarrer j. Each subcarrer j tends to be used by user, who has the best channel qualty,.e., large c,j, whle user wll also wn a gan n terms of λ for usng subcarrer j. Through nspecton, we can solve the subproblem as follows: j = arg and let max c,j (1 + λ ); x,j = { 1; f = j 0; otherwse j =(1, 2,..., K). The complexty of solvng all the K subproblems above s o(k) n contrast to the NP-hard complexty of the orgnal problem, whch ndcates that the above algorthm s very effcent. 2) Subgradent Based Soluton of Dual Problem: Snce the subproblems nvolves dscrete varables, the objectve functon f LD defned n Lagrange dual problem LD s concave, pecewse lnear and may not be dfferentable at certan pont n the λ space. The subgradent method s commonly used to solve ths knd of optmzaton problem. A subgradent method fnds the optmal soluton of f LD (λ) by teratve method n whch startng wth some λ 0, a sequence of λ n, whch eventually converges to the optmal soluton, s constructed accordng to: λ n+1 = λ n + Q n S n. (8) where n s the teraton ndex, Q n 0 s a sutable step length and S n s the subgradent vector of f LR (λ) at λ n.wehave: S n = cx n ˆr mn. (9) where x n s a soluton of problem SP defned above gven λ n, ˆr mn =(ˆr 1 mn, ˆr 2 mn,..., ˆr M mn ) and c s the achevable data rate matrx. We have: Q n =, and Q n 0, n. n=1 In order to get the optmal soluton, we may need an nfnte number of teratons, so to obtan a reasonable suboptmal soluton as quckly as possble, we defne Q n as follows: Q n = Q 0 ρ n, 0 <ρ<1. (10) where Q 0 s a ntal value and ρ s a system parameter. Q n decreases n exponental speed when n ncreases, so that the number of teratons s reduced. The updatng process stops Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE GLOBECOM 2006 proceedngs.
when the change of λ s small enough,.e., λ<εwhere ε s a predefned system parameter. 3) Obtanng Feasble Soluton: Snce the mnmum data rate constrants have been relaxed, the soluton of the dual problem above may be nfeasble,.e., constrans (2c) may not be met. We present a heurstc to convert the nfeasble subcarrer allocaton result to a feasble one. The man dea of our heurstc s to do the adjustment for each subcarrer, such that t leads to the least decrease n channel effcency. We call the heurstc Least Lose Frst (LLF), whch works as follows: Step 1: Let SC= the set of all subcarrers. Step 2: Pck a subcarrer j randomly from SC, Let be the MS that subcarrer j s allocate to,.e., x,j =1and U be the remanng. Step 3: If there s MS, where U and r + < ˆr mn,we select a MS accordng to: = arg mn(λ ). U Step 4: Assgn subcarrer j to MS and do: x,j =0, and x,j =1; r + = r, and r+ =(1 µ)r + µc,j. where µ s a parameter defned n next subsecton. Step 5: Set SC=SC\j. IfSC = φ, stop, else go to Step 2. The selecton of MS n step 2 comes from the dea that users wth small λ may have relatvely better channel qualty, so that the exchange may result n the least decrease n channel effcency. B. Servce Trackng and Overall Algorthm The above algorthm reles on the avalablty of ˆr mn for users at the begnnng of each subcarrer allocaton cycle. ˆr mn s calculated based on the hstory nformaton of subcarrer allocaton and the QoS requrements of users n terms of R mn. Next we provde a method to calculate ˆr mn based on the commonly used sldng wndow mechansm. We use T to stand for the sze of the sldng wndow n terms of tme slots and we have: r + = T 1 T r + 1 T ˆr. (11) where ˆr s the data rate receved n current subcarrer allocaton cycle. In order to guarantee the mnmum data rate requrements of users, we have: r + R mn. Let µ = 1 T, then we have: (1 µ)r + µˆr R mn ˆr mn tme ˆr mn ˆr mn = ˆr Rmn (1 µ)r µ. (12) equals to the mnmum value of ˆr, whle at the same must be larger than zero, so we have: 0 ; f R mn (1 µ)r < 0 R mn (1 µ)r µ ; otherwse (13) Based on the Lagrangan relaxaton subcarrer allocaton framework above, we present the overall algorthm,.e., LRSA as follows: Step 1: Choose a startng pont λ 0, Q 0, and let n=0; Step 2: Calculate ˆr mn accordng to (13) for all users, and solve the subproblem SP(λ n )by: f LR (λ n )=max { c,j (1 + λ n )x,j λ n ˆr mn }. =1 =1 Let x n be the soluton of ths subproblem. Step 3: Calculate subgradent vector S n at λ n by S n = cx n ˆr mn. Step 4: If S n 0, λ n s the optmal soluton then stop, otherwse go to Step 5. Step 5: If stop condton λ<εs met, then stop, otherwse go to Step 6. Step 6: Let λ n+1 = max {λ n + Q n S n, 0}, Q n = Q 0 ρ n, n=n+1, gotostep2. Step 7: Obtan the feasble soluton followng the LLF algorthm presented earler above. IV. SIMULATION RESULTS In ths secton, we study performance of LRSA by smulatons. Our smulaton s based on the mplementaton of IEEE 802.16. We consder a sngle cell wth one Base Staton and a varyng number of Moble Statons. We assume that there are 32 subcarrers (subchannels) n the wreless channel between the Base Staton and the Moble Statons. The frequency selectve fadng wreless channels are emulated by a nne-state Markov chan [9]. Dfferent states represent dfferent channel qualtes and data rates by whch users can transmt ther data. In ths paper, we are concerned wth data servces, that have prescrbed mnmum data rates requrements as defned n the IEEE 802.16 standard. A. Channel Effcency Fgure 1 shows channel effcency as functon of the number of users. We compare the channel effcency of LRSA wth that of Best Channel Frst (BCF), Wong s algorthm [4] and SAMDRA [10]. BCF chooses the best channel qualty user on each subcarrer, so t s the optmal soluton n the sense of channel effcency. As shown n Fgure 1, channel effcency ncreases when the number of users ncreases. Ths s because frequency dversty ncreases when more users enter nto the network, so that each subcarrer can be allocated to relatvely better channel qualty users. Compared to the other algorthms, Wong s algorthm has the lowest channel effcency. Ths s because Wong s algorthm restrcts the maxmum number of sucarrers that can be allocated to a user durng each allocaton cycle n order to guarantee the weghted proportonal farness among the users. We also can see that channel effcency of SAMDRA and LRSA s very hgh and s almost equal to that of the BCF Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE GLOBECOM 2006 proceedngs.
Channel Effcency 0.8 0.7 0.6 0.5 0.4 0.3 BCF SAMDRA-0.001 Wong's LRSA-0.1 LRSA-0.01 8 13 18 23 28 33 38 User Number Fg. 1: Channel Effcency Improvement vs System Payload optmal soluton when the number of admtted users s less than 25. On the other hand, channel effcency mprovement decreases when the number of user s more than 25, for at ths moment, the system becomes overloaded and the subcarrer allocaton algorthm must guarantee bad channel qualty users the mnmum data rates at the expense of channel effcency. LRSA has a hgher channel effcency compared to SAMDRA especally when the number of users s larger than 30. Fgure 1 also shows the effect of system parameter ε n LRSA, whch reflects the trade-off between channel effcency mprovement and computng complexty. B. Mnmum Data Rate Guarantee Fgure 2 shows the average data rates receved by users where users have dfferent channel qualtes. There are 30 users n total n the system. Channel qualty decreases as the user ndex ncreases from user 1 to user 6. The 24 other users have mddle level channel qualty as that of user 3. We can see that LRSA can allocate more subcarrers to better channel qualty users than the other algorthms, whle guaranteeng ther mnmum data rates. Ths s the reason why LRSA has hgh channel effcency as shown n Fgure 1. In Fgure 3, we assgn user 1, user 2 and user 3 wth dfferent channel qualtes,.e., user 1 wth hgh channel qualty, user 2 wth mddle and user 3 wth low. The three users have ther mnmum data rate requrement set to 30kbps (480 packet per second) 20kbps (320 packet per second) and 10kbps (160 packet per second), respectvely. As ndcated n Fgure 2, when the number of users ncreases, the date rates of user 1 user 2 and 3 decrease to the mnmum data rate prescrbed. When the system payolad s ncreasng, data rates of bad channel qualty users decrease to ther mnmum data rates earler. Ths s because LRSA s more lkely to allocate the subcarrers to users wth better channel qualtes n order to mprove the overall channel effcency. V. CONCLUSION In ths paper, we have consdered the subcarrer allocaton problem n OFDMA wreless channels. A practcally effcent Lagrangan Relaxaton based Subcarrer Allocaton (LRSA) scheme s proposed. LRSA s proven to be of polynomal complexty and provdes bounds on the value of the objectve functon,.e., channel effcency. Numercal results show that, n comparson wth other algorthms, LRSA can result n sgnfcant mprovement n channel effcency whle at the same tme guaranteeng the mnmum data rate of users. 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