Adaptive Resource Allocation in Multiuser OFDM Systems

Transcription

1 Adaptive Resource Allocatio i Multiuser OFDM Systems Fial Report Multidimesioal Digital Sigal Processig Malik Meherali Saleh The Uiversity of Texas at Austi malikmsaleh@mail.utexas.edu Sprig 005 Abstract I a sigle user commuicatio system, OFDM proves to be a robust techique to mitigate the ISI experieced o frequecy selective widebad wireless chaels. OFDM is suggested as a approach to be used i multiuser systems, particularly the dowlik sectio of a cellular system. The users of such a system must share resources such as trasmit power ad subcarriers. I order to icrease system performace, it is show that these resources should be allocated adaptively to users based o their chael iformatio. This report studies two adaptive allocatio approaches that maximize system capacity. Durig the implemetatio of these approaches, it was foud that sigificat complexity of the algorithms could be reduced by preferrig a equal power allocatio over the more widely studied water-fillig power allocatio. I. Itroductio ext geeratio wireless commuicatio systems (e.g. Fourth Geeratio 4G systems) aim to support iteractive multimedia service ad wireless Iteret access at a ambitious date rate of 00Mbps or more. High data rate commuicatio over widebad chaels is sigificatly limited by iter-symbol iterferece (ISI) [], which is a result of multiple copies of the trasmitted sigal created by the reflectio off objects (such as buildig ad cars). This pheomeo is commoly kow as multipath fadig. To combat ISI,

2 multicarrier modulatio techiques, icludig Orthogoal Frequecy Divisio Multiplexig (OFDM) are amog the possible solutios that have bee suggested. OFDM [] divides a broadbad chael ito arrow subcarriers of equal width such that the chael frequecy respose o a particular subcarrier is approximately flat. Appedig a guard bad or cyclic prefix (CP) to the trasmitted symbols coverts the liear covolutio of the symbols with the chael ito a circular oe, provided the legth of the CP is greater tha or equal to the chael legth (delay spread). Thus, each subcarrier ca be modeled as a gai (time ivariat or variat) with additive white Gaussia oise (AWG). This setup allows the received sigal to be ISI free ad modeled by the followig equatio y [ k] = H[ k] s[ k] + v[ k] () where y [k] is the received symbol, H [k] is the discrete Fourier trasform of the chael respose, s [k] is the trasmitted symbol ad v[k ] is AWG. II. Backgroud OFDM ca be icorporated ito a multiuser system by sedig multiple users data o differet subcarriers sice, the subcarriers are orthogoal to each other. A example of such a system could be a cellular system, particularly the dowlik, where the base statio trasmits to multiple users. Oe of the major tasks for such a system would be to allocate subcarriers, bits ad trasmissio power to the differet users. There are two approaches to allocate these resources: fixed ad adaptive. Fixed allocatio approach uses time divisio multiple access (TDMA) or frequecy divisio multiple access (FDMA) to allocate predetermied time slots or subcarriers to users as explaied i [4]. Fixed allocatio approach is ot a optimal approach to share the resources sice it does ot accout for the curret wireless chael state betwee the trasmitter ad user. A wireless chael is frequecy selective if there exists multipath fadig betwee the trasmitter ad receiver. Depedig o the surroudigs of each user at its locatio, there are diverse chael patters

3 betwee the base statio ad the users that vary with time. The chael diversity, created by multiple users, ca be used for adaptive allocatio of resources to icrease the system performace. The adaptive resource allocatio approach ca be formulated i two differet optimizatio techiques. The first kow as margi adaptive [5], miimizes the total trasmit power while maitaiig a fixed bit rate for each user. The secod also called rate adaptive [6, 7], maximizes the summed bit rate for all users (also kow as system capacity,) while keepig the trasmissio power costat. Sice power is a restricted ad limited quatity i actual systems, this report is goig to focus o the rate adaptive approach. For adaptive allocatio of resources the chael respose for each user eeds to be updated frequetly at the base statio. This icreases the burde o the base statio. It turs out that the complexity of the task is further icreased by the o-liear ature of these optimizatio problems. Oe way to reduce complexity is to covert the all variables to itegers [9]. evertheless, the complexity rises expoetially i the iteger programmig solutio as the umber of variables ad costraits icreases. The rate adaptive techiques metioed i Jag ad Lee i [6] maximizes the system capacity by allocatig each subcarrier to a user which has the best chael gai for it. This techique is ufair sice it eglects users which cosistetly see low chael gais. Such users may ever be assiged ay subcarrier if their chael persists to be deeply faded. O the other had, [7] maximizes the system rate while maitaiig the same data rate for all users all the time. Such a techique may ot be helpful for a system which requires differet data rates for users depedig o their service level agreemet with their system. To solve this issue authors i [8] have proposed a system which looks at proportioal fairess amogst the users data rates. A set of predetermied proportio data rates is maitaied betwee users while maximizig system capacity. This method is further modified i [] to give a low complexity optimizatio solutio ad is discussed later i the report.

4 The rest of this report is divided ito the followig sectios. Sectio III describes the system model for a rate adaptive scheme ad Sectio IV metios how [6,7, ] solve the optimizatio problem. Sectio V gives a descriptio of the implemetatio ad Sectio VI outlies the complexity issues i each of the optimizatio problems. Sectio VII discusses the simulatio results for two of the optimizatio problems ad Sectio VIII cocludes this report. III. System Model Sice [6, 7] both maximize the system capacity with total power as a costrait, they share the same geeral system model for the optimizatio problem. Followig is the mathematical represetatio of the system Subject to: p k, k, k= k= = ρ ρ p 0 = k, k, for all k { 0,} P total, for all k, = for all k, R = max pk,, ρk, k= = ρ IV. Algorithms k, log + pk, h B 0 k, Where, is total umber of users is total umber of Subcarriers o is power spectral desity of AWG B is total badwidth, Ptotal is total power Pk, is the power for kth user ad th subcarrier Pk, idicates if th subcarrier is used by kth user each subcarrier ca be used by oe user () The algorithms for Jag ad Lee [6], ad She et al. [7] are discussed i this sectio. To make the optimizatio problem tractable, Jag ad Lee divide the problem ito two stages: subcarrier allocatio ad power allocatio. They derived a theorem, which states that i order to maximize the data rate o a subcarrier, that subcarrier should be assiged to oly oe user who has the best chael gai for it. Followig the theorem the model further simplifies ito the followig equatio R = = log + p k * 0 h k B * (3)

5 = k for =,,, ad = * where k arg max{ h,, h,,..., h, } p k * = P total The system i (3) ca be treated as sigle user OFDM system ad power is allocated either usig waterfillig algorithm or authors propose equal power allocatio strategy to reduce complexity. Followig are the summarized steps for the algorithm O the other had, She et al. i [7] itroduces the proportioality fairess amog users by addig a oliear costraits give by, R = γ :...: γ (4) : R :...: R : γ where Rk is the k th user s data rate, ad { } i i= γ are the predetermied data rate ratio specified by the system. I order to reduce complexity i the biary ad iteger optimizatio problem (3) with oliear costrait (4) ad i fidig the global solutio for the system capacity over all possible subcarrier allocatio (where is total users ad is total subcarriers), the authors propose the suboptimal method; subcarrier allocatio followed by power allocatio. For subcarrier allocatio, the authors discuss the followig algorithm, which they base o [8]

6 For the power allocatio, the authors use the Lagragia optimizatio techique to geerate a set of oliear equatios that require iterative methods such as ewto-raphso or Quasi-ewto to obtai a solutio. To reduce this complexity, the authors metio cases where the equatios ca solved i oe iteratio by takig some assumptio. For the liear case, the assumptio is : :...: = γ : γ :...: γ (5) where i is the umber of subcarriers allocated to i th user. Wog et al. i [] combies this liear approach with a approach i [0] to reduce complexity sigificatly i the subcarrier ad power allocatio scheme. V. Implemetatio ad ew fidigs Wog et al i [] have further modified [7] to reduce the umber of operatios ad make the cocept of proportioal rates more feasible for real-time systems. This method will be further addressed as LIEAR. The implemetatio of [] is at this address: For compariso purpose, the High Chael-to-oise Ratio Case suggested by She et al. i [7] is icluded i the implemetatio ad will be further addressed as ROOTFIDIG. To compare Jag ad Lee method with LIEAR ad ROOTFIDG approach, I have developed a module i MATLAB which implemets the basic idea proposed by Jag ad Lee. My module uses the same chael fadig parameters, badwidth, total trasmit power ad various other variables used by Wog et al. i []. The module first assigs subcarriers accordig to Step defied i Sectio IV. I the ext stage power is allocated based o the two methods i the Step. The first method assumes the system is a sigle user OFDM system after the allocatio of subcarriers ad water-fillig algorithm is used to allocate power. For the secod method, I assume equal power o all subcarriers. After ruig the simulatio for my module (Jag ad Lee approach) ad observig the results, it appears that there may be margial performace differece betwee water-fillig ad equal power allocatio as see i Fig. i the simulatio Sectio VII. This result ca be used to save valuable computatio time which the water-fillig algorithm requires to iterate to the best solutio. I propose that this iterative procedure ca be replaced by the equal power allocatio scheme i [] to reduce iterative

7 computatios ad processig cost ad will be further addressed i the simulatio as LIEAR-WF for waterfillig ad LIEAR-EQ for equal power. VI. Complexity Compariso The computatioal complexity of the Jag ad Lee algorithm i [6] eed to be studied. Please ote that refers to the total umber of users i the system ad refers to the umber of subcarriers. Step a of the algorithm by Jag ad Lee requires costat time for iitializatio. Step b ivolves sortig th subcarrier gai for all users for every subcarrier, requirig ( ) O log operatios. Step ivolves the choice betwee two methods. Water-fillig method requires fiite umber of operatios ad the equal power method requires lesser umber of operatios. The complexity for [7,] is listed i the followig table. Table : Complexity Comparisio Table Steps LIEAR ROOTFIDIG Jag ad Lee log Subcarrier Allocatio O ( log ) O( log ) O( ) Power Allocatio O ( ), 9 O ( ) VII. Simulatio Results Simulatio parameters: Simulatio parameters for this paper were the same oes used i [] ad listed Badwidth Total Subcarriers BER Maximum delay spread SR Gap Total Power MHz 64 0^-3 5usecs W 0 differet chael realizatios were used durig 6 simulatio ad each realizatio was sampled times for each user. Proportioality ratios were assiged to users at each chael realizatio ad are the same as described i []. z ) /H y (bit/s pa ci t a c LIEAR-WF ROOT-FIDIG Jag ad Lee LIEAR-EQ Overall Capacity: Fig. displays the capacity 3.5 achieved by methods proposed by Jag ad Lee [6, umber of users Fig. 0: System Capacity versus umber of users. The capacity for Jag ad Lee is much higher

8 ad ]. The capacity achieved for the modified LIEAR, usig equal power allocatio (LIEAR Gamma ROOT-FIDIG LIEAR-WF LIEAR-EQ Capacity Maximizatio EQ) istead of water-fillig algorithm is also plotted. From the figure, there is egligible performace loss for usig equal power allocatio. Jag ad Lee outperforms i achievig a much greater system o r ti P ropo R ate e d a l iz or m capacity. System capacity Vs SR is give i Fig5. SR is defied as Ptotal/B*o Proportioality: Fig. shows the ormalized capacity proportios for the users. The extra bars for LIEAR- WF ad LIEAR-EQ bars emphasize that there is egligible performace loss whe usig equal power allocatio scheme ad cosiderable reductio i complexity. The figure also shows how users of Jag ad Lee get ufair share of the system capacity. This poit ca be further oted i Fig.3 where oly user, 3 ad 6 are give share of the capacity for a radom chael realizatio because they had the best chael gai for that realizatio. o s r ti User umber (k) P ropo 0.4 R ate e d a l iz 0.3 m or Fig. : ormalized Rate Proportio for 6 users. LIEAR(WF & EQ) ad ROOTFIDIG methods give fair share of the system capacity to the users. LIEAR User umber (k) 0 - sapshot Gamma ROOT-FIDIG LIEAR-WF LIEAR-EQ Capacity Maximizatio Fig. 3: A sapshot of rate proportio for a chael realizatio. Jag ad Lee method distribute the capacity to user,4 ad 6 oly. LIEAR(WF, EQ) ad ROOTFIDIG share accordig to rate proportios Average CPU time compariso LIEAR-WF LIEAR-EQ Complexity: Fig4. shows the reductio i complexity whe equal power allocatio is used for the LIEAR method. ) (s e m t i U C P A ve Table : Performace comparisio Methods LIEAR-EQ ROOTFIDIG Jag ad Lee QoS Good Very Good ot Good umber of users Fig. 4: Averagc CPU time for MATLAB code to execute LIEAR with water-fillig power allocatio ad LIEAR with equal power distributio. Equal power takes less time

9 System Capacity Vs SR VIII. Coclusio OFDM represets a successful approach to mitigatio of ISI i wireless commuicatio. Multiuser OFDM systems create chael diversity that ca be exploited to icrease z /H t/s y (bi c i t C apa m S yste LIEAR-WF ROOT-FIDIG Jag ad Lee LIEAR-EQ system performace. Due to the varyig coditios of the chael betwee the trasmitter ad SR db Fig. 4: System Capacity versus SR receiver, it is essetial to adaptively allocate the recourse based o istataeous chael kowledge. This paper compared two rate adaptive approaches that allocate resources. The approach take by She et al. [7] (modified later by Wog et al. to reduce complexity) provides proportioal fairess to users by givig a fair share of the system capacity depedig o their predetermied capacity ratios. O the other had, the Jag ad Lee [6] approach is show to provide a much higher system capacity by allowig oly those users to trasmit o subcarriers which had better gais for them. Durig implemetatio, it was see that Wog et al. i [] could further reduce the complexity by usig equal power distributio istead of the water-fillig method without sigificat loss i performace Refereces: [] B. Sklar, Rayleigh fadig chaels i mobile digital commuicatio systems.i. characterizatio, IEEE Commuicatios Magazie, vol. 35, pp , July 997. [] J. A. C. Bigham, Multicarrier Modulatio for Data Trasmisso: a Idea whose Time Has Come, IEEE Commuicatios Magazie, vol. 8, o. 5, pp. 5-4, May 990. [3] T. M. Cover ad J. A. Thomas, Elemets of Iformatio Theory. ew York: Wiley, 99. [4] H. Rohlig ad R. Gruheid, Performace compariso of differet multiple access schemes for the dowlik of a OFDM commuicatio system, i Proc. IEEE Vehicular Techology Cof., Phoeix, AZ, pp , 997 [5] C. Y. Wog, R. S. Cheg,. B. Letaief, ad R. D. Murch, Multiuser OFDM with adaptive subcarrier, bit, ad power allocatio, IEEE Joural o Selected Areas i Commuicatios, vol. 7, pp , Oct [6] J. Jag ad. B. Lee, Trasmit Power Adaptatio for Multiuser OFDM Systems, IEEE Joural o Selected Areas i Commuicatios, vol., o., pp. 7-78, Feb [7] Z. She, J. G. Adrews, ad B. L. Evas, Adaptive Resource Allocatio i Multiuser OFDM Systems with Proportioal Fairess, IEEE Trasactios o Wireless Commuicatios, accepted for publicatio. [8] W. Rhee ad J. M. Cioffi, Icrease i capacity of multiuser OFDM system usig dyamic subchael allocatio, i Proc. IEEE Vehicular Techology Cof., Tokyo, Japa, pp , May 000. [9] I. im, H. Lee, B. im, Y. Lee, O the use of liear programmig for dyamic subchael ad bit allocatio i multiuser OFDM, i Global Telecommuicatios Coferece, vol. 6, pp , ov. 00. [0] H. Yi ad H. Liu, A Effciet Multiuser Loadig Algorithm for OFDM-based Broadbad Wireless Systems, i Proc. IEEE Global Telecommuicatios Coferece, vol., 000, pp [] D. G. Lueberger, Optimizatio by Vector Space Methods. ew York: Wiley, 969. [] I. C. Wog, Z. She, B. L. Evas, J. G. Adrews, A Low Complexity Algorithm for Proportioal Resource Allocatio i OFDMA Systems, i IEEE Iteratioal Sigal Processig Systems Workshop, pp. -6, Oct. 3-5, 004, accepted for publicatio.