8 JOURAL OF COUICATIOS VOL. 4 O. FERUARY 009 Resource Allocatio i Dowli IO-OFDA with Proportioal Fairess i Da ad Chi Chug o Departmet of Electrical ad Computer Egieerig atioal Uiversity of Sigapore Email:{dabi eleocc}@us.edu.sg Abstract A ew scheme for IO-OFDA (ultiple Iput ultiple Output - Orthogoal Frequecy Divisio ultiple Access) dowli resource allocatio is preseted i this paper. It has a objective of maximizig the total system capacity while havig costraits o the total available power ad proportioal fairess. Domiat eigechaels obtaied from IO state matrix are used to formulate this scheme with low complexity. It gives a tradeoff strategy amog system capacity computatioal complexity ad proportioal fairess. Simulatio results show that this ew adaptive allocatio scheme ca provide much better capacity gai tha static allocatio methods while achieve ear optimal fairess of rate distributio with a liear complexity. As a result it may be suitable for fulfillig diverse QoS requiremets of IO-OFDA systems. Idex Te adaptive resource allocatio OFDA diverse QoS requiremets SISO/IO- I. ITRODUCTIO I recet years OFD (Orthogoal Frequecy Divisio ultiplex) as a promisig techique to fulfill the eed of various high data-rate trasmissio systems has bee extesively explored. This techique has the ability to combat multipath fadig shadowig path loss ad other time-varyig detrimetal effects i wireless eviromets ad is also robust agaist frequecy selective fadig ad iter-symbol iterferece (ISI). esides OFD is ofte exploited as a multiple access techique i high date-rate systems such as WLA ad Wiax. With joit bit power ad subcarrier allocatios ivestigated i [] OFDA ca exploit both frequecydomai diversity ad multiuser diversity to improve the spectrum efficiecy. IO (ultiple-iput ultiple-output) as aother promisig method ca greatly improve the physical layer performace of moder wireless commuicatio systems. ay studies [-3 9-0] of such systems have bee doe i multi-user eviromets. I IO systems multiple ateas are employed at both the trasmitter ad receiver to itroduce spatial diversity. With the eed to use additioal ateas IO receivers are geerally more complex ad are sometimes combied with OFD or OFDA to hadle the problems iduced by multi-path chael more efficietly. Specifically IO-OFDA has bee icorporated ito the IEEE 80.6e stadard ad IO-OFD is also recommeded i the IEEE 80. stadard. ost resource allocatio problems i SISO/IO- OFDA systems ivolve the optimizatio of a certai objective fuctio from two agles. Oe is to miimize the total trasmit power at the ase Statio (S) [-3] while aother oe is to maximize the total system capacity [4-]. Specifically the maximizatio of system capacity has bee studied i [4] without cosiderig rate fairess amog users for dowli SISO-OFDA systems. The this was exteded by She i [5] to achieve almost ideal proportioal rate distributio. For such a system Wog formulated a lower complexity algorithm i [6] ad Hui itroduced a priority based sequetial schedulig criteria to ehace system capacity with largely reduced fairess i [7]. I additio the tradeoff betwee capacity ad fairess has bee discussed i [8]. Due to differet atures betwee SISO ad IO systems the re-formulatio of SISO-OFDA resource allocatio scheme to IO-OFDA systems have also bee ivestigated i [-3 9-]. evertheless these algorithms seldom cosider rate fairess amog users or do ot have a flexible cotrollability o rate fairess. I the future high data-rate wireless commuicatio systems must dramatically icrease the system capacity to serve a large umber of users. Also the icreased capacity must be distributed to idividual users i a fair maer to guaratee diverse QoS requiremets []. Essetially capacity ehacemet ad fairess improvemet which are usually coflictig i ature are two crucial issues i resource allocatio. Specifically spectrum efficiecy i wireless systems is traditioally evaluated i te of total system capacity which will usually lead to ufair allocatio due to differet user chael coditios eve though the optimal capacity ca be guarateed []. O the other had absolute fairess may lead to low spectrum efficiecy. Clearly it will be desirable i may applicatios to have a tradeoff strategy betwee capacity ad fairess for allocatig wireless resources. 009 ACADEY PULISHER
JOURAL OF COUICATIOS VOL. 4 O. FERUARY 009 9 With these i mid the problem of dowli resource allocatio for IO-OFDA systems is ivestigated i this paper. It has a objective of maximizig the total system capacity subject to the total available power ad proportioal fairess. Especially the proportioal fairess is emphasized i a cotrollable maer by a iterative process. First a suboptimal power allocatio is discussed. The a ew scheme with low complexity is proposed to achieve cotrollability o capacity ad fairess. I this scheme a priority based schedulig method is cosidered to ehace fairess first ad the a Tradeoff-Factor (TF) is itroduced to rearrage subcarriers amog users with the aim of achievig least capacity loss ad most fairess gai uder the itroduced algorithm desig criteria. With this aim the ew scheme ca achieve ear optimal proportioal fairess amog users after maximum subcarrier rearragemets at a liear complexity. The rest of this paper is orgaized as follows. I sectio II the IO-OFDA system model ad the capacity optimizatio objective fuctio are itroduced. A suboptimal power allocatio method is discussed i sectio III. The the proposed scheme is described i detail i sectio IV. Sectio V shows some simulatio results ad Sectio VI cocludes this paper. Fig. Dowli IO-OFDA system diagram II. SYSTE ODEL Followig the implemetatio ad descriptio i [3-4 ] Fig. shows the bloc diagram of a dowli IO-OFDA system with a emphasis o resource allocatio. Without loss of geerality cosider a system with users ad subcarriers. Assume that each user has receivig ateas ad the base statio has r t trasmittig ateas. For user (= ) o subcarrier (= ) the chael state matrix is H with dimesio r. H ca be decomposed t through sigular value decompositio (SVD) as H H U V u v i H () where mi( r t ) is the ra of H { } are i the sigular values of H i descedig order { u } i ad { v } are the correspodig left ad right sigular i () vectors respectively. The maximum sigular value is geerally much larger tha the others { } i a i outdoor eviromet [3]. For coveiece the () decoupled SISO chael deoted by will be referred to as the domiat eige-chael for user o subcarrier. Fig. also shows how the resource allocatio scheme ca be performed. The S first uses the istataeous feedbac CSI to mae resource allocatio decisio. This decisio is forwarded to the IO-OFD trasmissio bloc. The the trasmitter loads each user s data oto its allocated subcarriers. It is assumed that the resource allocatio decisio is set to each user via a separate chael. Therefore each user ca decode the data o its assiged subcarriers. The resource allocatio scheme will be updated as fast as the chael iformatio is collected [5]. I [] where the optimality of IO- OFDA has bee ivestigated Li shows that oe subcarrier should be allocated to oly oe user at ay time i order to achieve optimality. ased o these studies the followig assumptios will be used i this paper: dowli IO fadig. varyig frequecy selective fadig chael which meas that the IO chael is assumed to be costat durig the subcarrier ad power allocatio process. that oly the sigular values of the chael state matrix will be required i the proposed resource allocatio scheme. ased o the optimizatio problem studied i [9-] the objective fuctio ivestigated i this paper ca be formulated as follows () max i log ( p g ) p () i subject to C: { } with { } beig disjoit sets C : p P with p 0 tot C3: R: R : : R : : : ote that the chael-to-oise gai of the ith eigechael for user o subcarrier is defied as g [ ] ad 0 p g gives the correspodig SR where 0 is the oise power. is the total available badwidth while the allocated power of user o subcarrier is ad the total available power is p P. The predetermied values tot { } are used to esure the proportioalities amog users i.e. / for =. } are disjoit sets { 009 ACADEY PULISHER
0 JOURAL OF COUICATIOS VOL. 4 O. FERUARY 009 where is the set of subcarriers assiged to user. The data rate of user is the R log p g (3) i ad the total data rate of all users is R tot R. I additio the domiat chael-to-oise gais of all users ad subcarriers ca be obtaied as () g [ ] (4) 0 for = ad =. The resource optimizatio problem give by () is P-hard implyig that it is difficult to obtai a solutio withi ay reasoable time frame. As a result a ew scheme is formulated shortly to solve this problem with liear complexity. III. POWER ALLOCATIO I this sectio the power allocatio is briefly discussed. As show i [] the optimal power ad subcarrier allocatio are iterleaved. Thus it is computatioally prohibitive to solve the optimal power allocatio. As a result some suboptimal power allocatio methods have bee proposed i [9-0 ] to achieve a performace that is a little worse tha optimality. The easiest way is to use equal power allocatio across all subcarriers uder differet access protocols. The suitability of equal power allocatio i our study is show as follows. With subcarrier allocatio havig bee carried out i.e. { } have bee determied the problem give by () is equivalet to maximizig the followig cost fuctio by usig Lagragia relaxatio. L log p g p Ptot i log p g log p g i i (5) Differetiatig L with respect to p ad settig each derivative to 0 we ca obtai the optimal power distributio for a sigle user g g m (6) i p g i p mg m for m m ad =. ote that the fuctio f ( x) x /( px) is mootoically icreasig ad teds to be p i.e. lim [ x /( px)] p. Thus a x approximatio to (6) ca be obtaied uder high SR as follows. (7) p p m That is equivalet to p p for m m m ad =. Due to o subcarrier sharig amog users it ca be cocluded that power could be approximately distributed across all subcarriers of each user i a equal maer. Obviously this equal power allocatio is ot optimized to improve fairess. However it ca achieve ear optimal proportioal fairess for our ivestigated problem whe beig combied with the proposed subcarrier allocatio which will show i the subsequet sectio. IV. PROPOSED SCHEE A ew scheme which ca maximize system capacity ad ehace proportioal rate fairess i a cotrollable maer is described i detail i this sectio. ased o IO chael state iformatio domiat eigechaels with gais i (4) are used to determie subcarrier allocatio with low complexity. I additio a Tradeoff Factor (TF) used to rearrage the subcarriers amog users is itroduced i the proposed scheme. This itroduced TF ca iteratively achieve ear optimal proportioal fairess amog users at the cost of certai reduced system capacity ad icreased complexity. To reduce complexity this scheme cosists of two separate stages which are subcarrier allocatio ad power allocatio as i [-3 0]. The five-step scheme is described i detail as follows ad its performace will be studied later. Step Parameter Iitializatio { }; { }; R 0; p P / ; for ; tot for ; u ; c 0 for ; where ad are the sets of user ad subcarrier idices R is the data rate of user represets the set of subcarriers allocated to user idicates the umber of eige-chaels eeded for user is the umber of u uallocated subcarriers p deotes equal power allocatio across all eige-chaels ad c is the subcarrier allocatio idicator ad c if subcarrier is allocated to user. Step First Subcarrier Allocatio I this step each user will be allocated its first subcarrier uder the schedulig priciple that the user with less pre-assiged umber of eige-chaels has higher priority to choose its first subcarrier. Clearly such user will be protected to select a subcarrier with better chael coditios ad teds to achieve higher capacity. Rate fairess ad system capacity are thus iitially emphasized. The subcarrier selectio i this step is doe by usig the domiat chael-to-oise gais i (4). That is if g is selected for certai ad the set { g } to i which g belogs will cotribute to update the istataeous rate (3) of user. The details are show as follows. 009 ACADEY PULISHER
JOURAL OF COUICATIOS VOL. 4 O. FERUARY 009 For j= to Do arg mi ; arg max g ; {}; c ; {}; {}; ; R R i log ( pg Ed For where {v} meas deletig elemet v from set A ad {v} stads for addig elemet v ito set A. After doig this step subcarriers have bee chose ad. Thus there are (-) subcarriers available for all users to select ad should be re-iitialized i the subsequet step. Step Priority-based Subcarrier Allocatio Proportioal fairess amog users is emphasized i this step. The basic allocatio priciple here is that the user with the least ratio of istataeously achieved rate to its required proportio has the priority to choose oe subcarrier at a time. Obviously the user with most starvatio to its desired proportio ca feed itself as much as possible through choosig additioal subcarriers with better chael coditios. This step is described as follows. { }; While U Do u arg mir ; If 0 Do arg max g ; ); {}; c ; {}; ; R R Else }; i log ( pg { Ed If Ed While ote that deotes the umber of elemets of set ad R is the ratio of the istataeously achieved rate to the required proportio of user. After this step 0 for ad. Step Residual Subcarrier Allocatio I this fial step a additioally simple procedure is doe that is all residual subcarriers are allocated to users with the objective of ehacig the total system capacity. { }; For choose the fist subcarrier idex from Do arg max g ; {}; {}; c ; {}; ); R R i log ( pg Ed For Step Subcarrier Rearragemet After previous four steps the maximum system capacity uder our scheme is achieved. Subsequetly a ovel step to rearrage subcarriers amog user with most fairess gai ad least capacity loss is used. The followig D value which is the root mea square of is defied to compare proportioal fairess amog users. D (8) where R R is the ormalized practical rate tot proportio of user. The value i (8) ca show the overall proportio derivatio of all users from their desired proportios. Whe this value becomes smaller the proportioal fairess would be ehaced. I the extreme absolute fairess is achieved whe D 0. A Tradeoff-Factor (TF) is used to cotrol the umber of subcarrier exchages ad details of this process are show as follows. While TF>0 Do {... }; {... }; R tot R s arg mi R / R ; s s tot s barg max R / R ; b b tot b ); D old / R R ; tot R t Rs; Rt Rb; earg mi( c g c g ) for e0; b b s s () i s s log ( s e); i () i b b log ( b e); i R R pg R R pg D ew / R R ; tot ew old If D D Do c 0; c ; be se Else Rs Rt; Rb Rt; brea; TF=TF-; Ed If Ed While where e is selected to esure least capacity loss. I each loop subcarriers betwee two selected users with most ufair proportios are exchaged. Thus this step is a iterative exchage process which ca ehace fairess at the cost of losig certai amout of system capacity. Although three or more users could be selected at oe time i this step however much more sophisticated subcarrier exchage method should be used. To reduce complexity two most ufair users are thus selected at a time. To esure covergece this step will be stopped oce the proportioal fairess caot be improved. 009 ACADEY PULISHER
JOURAL OF COUICATIOS VOL. 4 O. FERUARY 009 Step Step Step Step Step TALE I Order of complexity O() O() O((-)) O() O(TF) The above five steps form our proposed scheme which ca compromise capacity fairess ad complexity i a cotrollable maer. Thus this scheme may be suitable for diverse QoS requiremets. I additio Table I briefly shows the computatioal complexity of each step. ote that is assumed to be much larger tha ad. V. SIULATIO RESULTS I the step the followig parameters are cofigured for computer simulatios. The frequecy selective multipath chael is modeled as cosistig of six multi-paths with a expoetially decayig profile for dowli IO chael betwee ay couple of trasmittig ad receivig ateas. A maximum delay spread is assumed to be 5s ad the maximum Doppler frequecy shift beig 30Hz. The total available power is assumed to be W ad the total badwidth is Hz. I additio the AWG power spectral desity is -80 dw/hz. To compare system capacity the proportioality costraits assiged to each user are assumed to have the followig probability fuctio i each chael realizatio. with probability 0.5 (9) p with probability 0.3 4 with probability 0. Static OFD-FDA ad OFD-TDA resource allocatio schemes will be preseted for compariso. Static OFD-FDA as a special case of adaptive OFDA allocates fixed sequeces of subcarriers to each user accordig to their proportioality costraits ad the allocated subcarriers of each user caot be chaged over time. I the followig statemets adaptive oly refers to o-static allocatio that meas subcarriers are expected to have differet allocatios over differet time slots. O the other had OFD-TDA allocates all spectrum resources to oly oe user at each time slot. The simulatio results are preseted i Fig. - 4 ad Table II. The umber of user varies from to 6 while 56 subcarriers ad r=t= are cosidered. Fig. depicts the total system capacity proportioality costraits give by (9). Obviously adaptive allocatio method ca outperform both FDA ad TDA methods. It ca be observed that the proposed scheme without subcarrier rearragemet i.e. TF=0 ca achieve the maximum capacity. I additio larger TF value will lead to more reduced capacity. ote that the system capacity uder this ew scheme icreases with the icreasig umber of users which is due to the multi-user Capacity (bit/s/hz)..8.6.4. 0.8 0.6 PropScheme TF=0 PropScheme TF=5 PropScheme TF=0 PropScheme TF=max FDA TDA 0.4 4 6 8 0 4 6 umber of Users Fig. = to 6 =56 r=t= total capacity vs. user umber. TF Values ormalized Rate Proportios Fig. 4 4 0 8 6 4 4 6 8 0 4 6 umber of Users 0.4 0. 0.0 0.08 0.06 0.04 0.0 0.00 Fig. 3 = to6 =56 r=t= TF value rage. Gamma PropScheme TF=max PropScheme TF=0 FDA TDA 3 4 5 6 7 8 9 0 3 4 5 6 User Idex =6 =56 r=t=ormalized rate ratios per user with the desired proportios (Gamma - the leftmost bar). diversity. evertheless both FDA ad TDA schemes do ot have such a property. Fig. 3 gives the TF value rage (upper rouded iteger) suggestig a maximum value i Step to cotrol the umber of subcarrier exchages. As see from this figure larger umber of users will result i more umber of subcarrier exchages due to the icreased multi-user diversity. Fig. 4 shows the ormalized proportios for 6 users uder the proportio- 009 ACADEY PULISHER
JOURAL OF COUICATIOS VOL. 4 O. FERUARY 009 3 TALE II Fairess compariso for differet schemes D i (8) (=6 =56 r=t=) Tau Set j 0 3 4 * j 4 8 6 PropScheme (TF=0) 0.0080 0.008 0.0084 0.009 0.0099 PropScheme (TF=5) 0.0054 0.0057 0.0057 0.0063 0.0069 PropScheme (TF=0) 0.0034 0.0037 0.0036 0.004 0.0045 PropScheme (TF=max) 0.0009 0.0009 0.0009 0.000 0.000 FDA 0.0094 0.0096 0.0096 0.000 0.04 TDA 0.0395 0.036 0.0364 0.0504 0.0767 *the other tau values are set to be 3 6 4 7 0 6 ality costrait of 3 4 4 8 9 6. This figure presets that the proposed scheme with maximum TF value (TF=4 i this simulatio) ca achieve ear optimal proportioal fairess. I additio Table II shows a detailed fairess compariso measured i (8) uder five differet proportioality costraits. As ca be see the proposed scheme ca give a iteratively refied performace as TF value icreases while the maximal umber of iteratios leads to the a almost ideal fairess. I additio the proposed scheme ca outperform traditioal FDA ad TDA methods. VI. COCLUSTIOS A ew adaptive resource allocatio scheme with flexible cotrollability o capacity fairess ad complexity is ivestigated i this paper. With above aalyses i mid some coclusios ca be draw. The proposed scheme with maximum TF value is suggested to be used i strict fairess sceario however it has most reduced system capacity. This ew scheme without subcarrier exchage may be used to capacity-emphasized sceario with acceptable reduced fairess. A wellselected TF value ca tradeoff betwee the previous two extreme. The flexibility of our proposed has bee demostrated i iformatio-theoretical viewpoit ad its implemetatio i practical IO-OFDA systems will be studied i further research. REFERECES [] C. Y. Wog R. S. Cheg.. Letaief ad R. D. urch ultiuser OFD with Adaptive Subcarrier it ad Power Allocatio IEEE J. Select. Areas Commu. vol. 7 pp. 747-758 Oct. 999. [] J.. Choi J.S. wa H. S. im ad J. H. Lee Adaptive Subcarrier it ad Power Allocatio Algorithm for IO- OFDA System i Proc. VTC 004-Sprig vol.3 pp.80 805. [3] Y. J. Zhag ad.. Letaief A Efficiet Resource-Allocatio Scheme for Spatial ultiuser Access i IO/OFD Systems IEEE Tras. Commu. vol.53 pp.07-6 Ja. 005. [4] W. Rhee ad J.. Cioffi Icrease i Capacity of ultiuser OFD System Usig Dyamic Subchael Allocatio i Proc. IEEE VTC Japa ay 000 pp. 085-089. [5] Z. She J. G. Adrews ad. L. Evas Adaptive Resource Allocatio i ultiuser OFD Systems With Proportioal Rate Costraits IEEE Tras. wireless commu. vol. 4 pp. 76-737 ov. 005. [6] I. C. Wog Zuag She. L. Evas ad J. G. Adrews A Low Complexity Algorithm for Proportioal Resource Allocatio i OFDA Systems i Proc. Sigal Processig Systems Cof. SIPS 004 pp. -6. [7] J. Hui ad Y. Zhou Ehaced Rate Adaptive Resource Allocatio Scheme i Dowli OFDA System i Proc. IEEE VTC006- Sprig vol. 5 pp. 464-468. [8] i Da C. C. o A ew Scheme with Cotrollable Capacity ad Fairess for OFDA Dowli Resource Allocatio i Proc. IEEE VTC 007-fall pp. 87-8. [9]. obayashi G. Caire Iterative Waterfillig for Weighted Rate Sum aximizatio i IO-OFD roadcast Chaels i Proc. ICASSP Cof. vol. 3 April 007 pp. 5-8. [0] J. Xu J. im W. Pai J. S. Seo Adaptive Resource Allocatio Algorithm with Fairess for IO-OFDA System i Proc.VTC006 Sprig vol. 4 pp. 585-589. [] G. Li ad H. Liu O the Optimality of Dowli OFDA IO Systems i Proc. Sigals Systems ad Computers Cof. vol. ov. 004 pp. 34-38. [].S. aw ad I. Sasase Resource Allocatio Scheme i IO- OFDA System for User s Differet Data Throughput Requiremets i Proc. IEEE WCC arch 007 pp. 706-70. i Da received the.sc. degree i 006 i electrical egieerig from Hohai Uiversity ajig Chia. He is curretly worig toward the Ph.D degree with the Departmet of Electrical ad Computer Egieerig atioal Uiversity of Sigapore. His curret research iterests focus o resource maagemet i multi-user wireless broadbad systems. Chi Chug o received the.sc. ad Ph.D. degrees i electrical egieerig from Loughborough Uiversity of Techology Leicestershire U.. He is curretly a Professor with the Departmet of Electrical ad Computer Egieerig atioal Uiversity of Sigapore. He serves as a Editor of the EURASIP Joural o Wireless Commuicatios ad etworig ad the ETRI Joural. His curret research iterests iclude digital sigal processig adaptive arrays commuicatios ad etwors. He is the author or coauthor of more tha 50 techical publicatios i the aforemetioed areas. Prof. o has served as a Associate Editor of the IEEE TRASACTIOS O SIGAL PROCESSIG ad the IEEE TRASACTIOS O ATEAS AD PROPAGATIO. 009 ACADEY PULISHER