2012, TextRoad Publicatio ISSN 2090-4304 Joural of Basic ad Applied Scietific Research www.textroad.com A Fuzzy Rule Base Assisted Adaptive Codig ad Modulatio Scheme for OFDM Systems Atta-ur-Rahma a,c,*, Ijaz Masoor Qureshi a,b,c, Aqdas Naveed Malik a,c a SEAS, Isra Uiversity, Islamabad Campus, Islamabad, Pakista b Air Uiversity, Islamabad, Pakista c Istitute of Sigals, Systems ad Softcomputig (ISSS) ABSTRACT Adaptive commuicatio is oe of the key techologies used to ehace the capabilities of future commuicatio systems. May adaptive bit ad power loadig techiques have bee ivestigated i the literature for ehacemet of trasmissio rate i combiatio with Orthogoal Frequecy Divisio Multiplexig (OFDM). I these systems maily either modulatio or codig scheme was adaptive but ot both. That is, oe of the two aspects was fixed while other was adaptive e.g., adaptive modulatio with fixed chael code or adaptive codig with a fixed modulatio alphabet. I this paper, we propose a ew scheme to adapt both code rate ad modulatio size by solvig a o-covex optimizatio problem usig a Fuzzy Rule Base System (FRBS) to ehace the achievable data rate i a OFDM system with a fixed target bit error rate ad fixed trasmit power for each subcarrier. Keywords: OFDM; FRBS; Adaptive Modulatio ad Codig; Modulatio Code Pair. I. INTRODUCTION Adaptive Orthogoal Frequecy Divisio Multiplexig is oe of the successful cadidates for may 3rd ad 4th Geeratio Systems. I this techique a sigle very high data stream is divided ito several low data rate streams. The these streams are modulated over differet orthogoal subcarriers. Cocept of adaptive commuicatio is though ot ew, however, it was cofied to adaptive modulatio oly i past decade. Combiig practical chael codig is quite obvious but mostly eglected due to aalytical limitatios. Adaptive modulatio for OFDM was proposed by Kelet [1] as early as 1989. Later same idea was examied for Gaussia slowly varyig dispersive fixed liks by Chow et al. [2] ad for widebad radio chaels by Cyzlwic [3] i 1996. Several researchers take adaptive modulatio as a optimizatio problem ad propose efficiet solutio for ecoded systems, like [4]. Fuzzy system based adaptive modulatio scheme for OFDM system was proposed by [5] i 2009. Turbo coded adaptive modulatio was ivestigated by Hazo et al [6] ad also differet adaptatio schemes were aalyzed [7]. For sigle atea OFDM systems, coded bit ad power loadig problem was addressed by Li et al. usig LDPC codes[8] origially motivated by [9]. May bit iterleaved coded modulatio (BICM) systems have bee proposed like [9, 10, 11]. Adaptive commuicatio usig turbo codes was also ivestigated i [12]. I may of the above schemes decisio for the ext trasmissio rate was based o SNR thresholds. Ad i most of the above coded systems code rate was fixed ad modulatio was adaptive. Similarly, i his doctoral dissertatio, Al-Askary [13] has proposed a adaptive OFDM system where modulatio for fixed for all subcarriers ad code rates were varyig over subcarriers usig cocateated codes. A adaptive codig ad modulatio scheme is proposed by Bockelma et al i which a bisectio method is used to adapt the trasmit rate [14]. This bisectio method was origially used by Krogold et al [15] to adapt power, modulatio ad code rate for a fixed chael. I this paper we proposed a Fuzzy Rule Base System to adapt the code rate ad modulatio scheme accordig to varyig chael coditios based upo set of rules that are derived from AWGN simulatio results of various code rates (covolutioal codes) ad modulatio schemes (QAM family). This techique is compared to other coded ad ecoded adaptive loadig systems to show sigificace of Fuzzy Rule Base System over others. The remaider of this paper is orgaized as follows. I sectio 2, system model is itroduced. Performace of differet codes i cojuctio with differet modulatios is preseted i sectio 3. The results of sectio 3 are used i sectio 4 to obtai a costraied optimizatio problem. I sectio 5 a brief itroductio to Fuzzy Rule Base is give that is used to solve the optimizatio problem formulated i previous sectio. Sectio 6 cotais the performace compariso of this scheme with various other famous adaptive schemes while sectio 7 cocludes the paper. *Correspodig Author: Atta-ur-Rahma, Postgraduate Research Studet, School of Egieerig & Applied Scieces (SEAS), ISRA Uiversity, Islamabad Campus, Islamabad, 46000, Pakista. Email: ataurahma@biit.edu.pk 4843
Atta-ur-Rahma et al., 2012 II. SYSTEM MODEL The system model cosidered is OFDM equivalet basebad model with N umber of subcarriers. It is assumed that complete chael state iformatio (CSI) is kow at both trasmitter ad receiver. The frequecy domai represetatio of system is give by r h. p. x z ; 1,2,..., N (1) where r, h, p, x ad z deote received sigal, chael coefficiet, trasmit amplitude, trasmit symbol ad the Gaussia oise of subcarrier 1, 2,..., N, respectively. The overall trasmit power of the system is T N P p Np sice power is same for all subcarriers, ad the oise distributio is complex Gaussia with zero 1 mea ad uit variace. It is assumed that sigal trasmitted o the kth subcarrier is propagated over a idepedet o-dispersive sigle-path Rayleigh Fadig chael ad where each subcarrier faces a differet amout of fadig idepedet of each other. Hece, the chael coefficiet of kth subcarrier ca be expressed as: j h e ; 1,2,..., N (2) where is Rayleigh distributed radom variable of th subcarrier, ad the phase is uiformly distributed over 0,2. A. Codig Schemes The codes used i this paper are o-recursive covolutioal codes with code rates take from the set C {1/ 4,1/ 3,1/ 2,2 / 3,3/ 4} with costrait legth 3. For decodig, stadard soft output Viterbi decoder is used. For coveiece, message bits legth is take equal to the umber of subcarriers ad stadard OFDM iterleaver/deiterleaver are used i simulatios. B. Modulatio Schemes I this paper we have utilized the symbols take from Quadrature Amplitude Modulatio (QAM), with rectagular costellatio. The modulatio symbols are take from the set M = {2,4,8,16,32,64,128} III. CODED MODULATION For experimetatio the sequece of operatios is carried out i same way as give i the figure1. The trasmitted sigal is first ecoded usig stadard feed-forward covolutioal ecoder havig code rate from the set C ad the the ecoded sigal is modulated usig the elemets of QAM from the set M. All codes used belog to set C with costrait legth 3 each ad modulatio symbols from set M with symbols from rectagular QAM costellatio. I this way we have followig pairs of codig ad modulatio by cross product of sets C ad M, which yields, P Cx M {( c, m ); c C, m M } (3) i j i j The graph for each pair is obtaied over a Additive White Gaussia Noise (AWGN) chael. The selectio of this chael is suitable i a sese that it reflects the proper relatioship betwee sigal to oise ratio (SNR) ad data rate achievable uder a specific target bit error rate (BER). Also other chael characteristics like fadig types etc ca be compesated easily. Graphs are depicted i the figures 1, 2, 3, 4 ad 5 with rate 1/4, 1/3, 1/2, 2/3 ad 3/5 respectively. All of the modulatio schemes are ivestigated with each code rate. 4844
Figure 1. BER compariso of differet QAM modulatios usig rate 1/4 covolutioal codes Figure 2. BER compariso of differet QAM modulatios usig rate 1/3 covolutioal code Figure 3. BER compariso of differet QAM modulatios usig rate 1/2 covolutioal code 4845
Atta-ur-Rahma et al., 2012 Figure 4. BER compariso of differet QAM modulatios usig rate 2/3 covolutioal code Figure 5. BER compariso of differet QAM modulatios usig rate 3/4 covolutioal code Adaptive system will make use of these facts ad it will use appropriate modulatio code pair (MCP) for each subcarrier depedig upo chael coditio at that subcarrier. That is over a subcarrier with good chael coditio a high code rate with a high QAM ca be used. If chael coditio is poor the low code rate with low QAM ca be used. Similarly the bled of differet MCP ca be used to cope up with etire spectrum of chael variatios. As far as our demostratio is cocered, all of the modulatio-code pairs are ivestigated, but obviously ot all pairs will be used. Oly a subset of these pairs will be cosidered due to the fact that two or more pairs may provide same throughput (modulatio code product) but with differet SNR demads. So, amog them, we will choose the pair with least SNR demad. For example, 16QAM with rate 1/4 ad 4QAM with rate 1/2 yields same throughput i.e. 1bit/sec/Hz but with differet SNR requiremets. So 4QAM with rate half will be chose due to less SNR demaded. Now the performace of the system will be characterized i subsequet sectios by usig the above results. IV. RATE OPTIMIZATION I order to maximize the rate for OFDM followig costraied optimizatio problem will be cosidered. N 1 max RTotal r N 1 s.t, (4) BER BER ad p P QoS T 1 N 4846
Where r (log 2 ( M )) RC, is bit rate of th subcarrier, which is product of code rate ad modulatio order used, P T is the available trasmit power ad BER QoS is target BER that depeds upo a specific quality of service (QoS) request or applicatio requiremet over th subcarrier, while N is umber of subcarriers i OFDM system. The power distributio is cosidered as equal for all the subcarriers. From the results obtaied i sectio-iii, those code-modulatio pairs that fulfill differet BER demads 5 4 3 2 depedig upo differet quality of services i.e. BER T 10,10,10,10 etc are obtaied. These are obtaied by drawig straight lies o the graphs show i figures 1 to 5, o certai BER poits (Quality of Service) 5 4 3 2 like 10,10,10,10. The the poits of itersectio of these lies ad the curves (represetig a code ad a modulatio) are oted that gives the appropriate SNR value. So the iformatio obtaied ca be expressed as for a give SNR ad specific QoS which modulatio code pair ca be used. I order to obtai more graularity BER poits are eve quatized. I this way there are about 500 pairs are obtaied from the graphs. Few of these pairs so called iput/output pairs (I/O pairs), are elisted i table 1. TABLE I. I/O PAIRS OBTAINED FROM SIMULATION SNR BER MCP SNR BER MCP 1.72 10e-2 [4,1/4] 1.529 10e-4 [2,1/3] 8.732 10e-2 [8,1/4] 5.093 10e-4 [4,1/3] 10.18 10e-2 [16,1/4] 12.25 10e-4 [8,1/3] 13.63 10e-2 [32,1/4] 13.59 10e-4 [16,1/3] 18.08 10e-2 [64,1/4] 16.98 10e-4 [32,1/3] 21.45 10e-2 [128,1/4] 21.63 10e-4 [64,1/3] 0.143 10e-2 [2,1/3] 24.27 10e-4 [128,1/3] 4.109 10e-3 [4,1/3] 17.62 10e-5 [32,1/3] 11.16 10e-3 [8,1/3] 22.68 10e-5 [64,1/3] 12.56 10e-3 [16,1/3] 24.83 10e-5 [128,1/3] This table after completio is used as a startig poit for geeratio of look-up table for the fuzzy rule base system. Without loss of geerality we ca say that this table represets a fuctio (mappig) i which the throughput ca be expressed i terms of BER ad SNR. R MCP f ( SNR, BER ) (5) The mappig show i eq-4 is a o-covex fuctio that caot be optimized usig covex optimal techiques uless it is made covex accordig to [14]. However, this fuctio is optimized by the proposed Fuzzy Rule Base System described i ext sectio. The steps ivolved i creatio of FRBS are described i the flowchart give i figure 6. The brief descriptio of each phase of the flowchart is give below. A. Graphs Graphs for differet combiatios of Codes ad Modulatio schemes are obtaied that are depicted i sectio-iii. B. Data Acquisitio Data is obtaied from the graphs i terms of iput/output (IO) pairs. This is take by puttig the horizotal lies for various bit error rates ad poits of itersectio with the curves are oted. C. Rule Formulatio Rules for each pair are obtaied by the appropriate fuzzy set used. That is by puttig complete pair i iput/output set ad a rule geerated for each pair. D. Elimiatio of Coflictig Rule The rules havig same IF part but differet THEN parts are kow as coflictig rules. This appears whe more tha oe modulatio code pair (MCP) are available for give specificatio. For istace, there is a rule whose THEN part cotais three differet MCP amely, [8,1/2], [16,2/3] ad [16,3/4]. Now [16,3/4] is best amog the rest sice its throughput is 4x3/4=3 while others have 3x1/2=1.5 ad 4x2/3=2.67 respectively. Similarly, sometime there could be two differet pairs with same throughput like [2,1/2] ad [4,1/4] both have same throughput that is 1x1/2=0.5, the [2,1/2] will be chose sice it exhibits less modulatio/demodulatio ad codig/decodig cost. E. Completio of Lookup Table Sice i lookup table scheme we may ot have complete umber of IO pairs, the those parts are filled by heuristic or expert kowledge. For example, a modulatio code pair is suggested by rule for a certai SNR ad QoS. The that rule ca also be used for slightly above SNR ad poor QoS. For istace, [128,3/4] is suggested 4847
Atta-ur-Rahma et al., 2012 3 for 25dB SNR ad BER 10 2 the this pair ca be used for 26-30dB SNR ad 10 BER cases as well. Sice if a modulatio code pair performs for lower SNR, the it ca easily sustai i higher SNR situatios. Similarly, if a MCP performs for a good QoS the it ca sustai for poor QoS demads. F. Fuzzy Rule Base Creatio Usig the Lookup table i above phase Fuzzy Rule Base is created usig Fuzzy Logic Toolbox i MATLAB. Further details are give i ext sectio. Figure 6. Fuzzy Rule Base System Creatio Flowchart V. FUZZY RULE BASE SYSTEM We propose a fuzzy rule base system (FRBS), which is capable of decidig the best modulatio code pair (MCP) for the ext trasmissio, based upo the heuristics. Fuzzy logic is best suited for the situatios that are vague, ambiguous, oisy or missig certai iformatio. Also fuzzy systems are very easy to implemet i hardware. This fuzzy rule base system (FRBS) is used to optimize the cost fuctio give i equ-3. It will be decided that which modulatio code pair is suitable for a specific subcarrier based upo the idividual chael state iformatio (CSI) at the subcarriers ad the Quality of Service demad. A table look-up scheme is used for desig of this fuzzy rule base system usig the followig steps. The iputoutput pairs eeded for desig of FRBS are provided i table-i. They are of the form; p p p ( x, x ; y ); p 1,2,3... M (6) 1 2 p p p Where x 1 represets received SNR, x 2 represets required BER (QoS) ad y represets the output MCP suggested by FRBS, so the rule format will be give as below; {IF ( x 1 is L1 ad x 2 is Q7) THEN y is P2} (7) Followig is the brief descriptio of differet compoets of fuzzy rule based system used. Desig of the FRBS is carried out i MATLAB 7.0 stadard Fuzzy System Toolbox. A. Fuzzy Sets Sufficiet umbers of fuzzy sets are used to cover the iput output spaces. There are two iput variables received SNR ad mius log bit error rate (MLBER) that represets a QoS. The reaso takig MLBER is 2 3 4 because BER of a required QoS is give by 10,10,10 etc while the rage of fuzzy variable should be equally spaced ad quatifiable. So to get this, followig operatio is doe first. MLBER log( BER ) BER 10 q q MLBER log(10 ) q There is oe output variable for modulatio code pair MCP. All of these iput ad output variables are depicted i figures 7, 8 ad 9 respectively. There are thirty-oe, sixtee ad twety-five fuzzy sets used for the variables SNR, MLBER ad MCP, respectively. B. Fuzzifier Stadard triagular fuzzifier is used with AND as MIN ad OR as MAX. These fuctios ca be see i figures 7, 8 ad 9 respectively. C. Rule Base Rule base cotais rules agaist all the IO pairs. As there are thirty-oe sets (L0 to L30) for first iput variable amed SNR ad about sixtee sets (Q1 to Q16) for iput variable MLBER. Hece there are 496 rules i rule base. Rule base is complete i a sese that rules are defied for all possible combiatios of iput space. (8) 4848
D. Iferece Egie Stadard Mamdai Iferece Egie (MIE) is used that will ifer which iput pair will be mapped o to which output poit. It is show i figures 10 ad 11. E. De-Fuzzifier Stadard Ceter Average Defuzzifier (CAD) is used for defuzzificatio. Figure 12 depicts the rule surface that shows that by icreasig SNR the throughput is maximized. Also o the other had for poor QoS throughput is more tha that of high QoS. A combied effect of both iput variables amely SNR ad QoS ca be see i that figure. For the highest value of SNR ad lowest value of QoS, throughput of the system approaches to 5bits/s/Hz. Figure 7. Membership fuctio of first iput variable SNR Figure 8. Membership fuctio of secod iput variable QoS Figure 9. Membership fuctio of output variable MCP 4849
Atta-ur-Rahma et al., 2012 Figure 10. Fuzzy Rule Editor Figure 11. Fuzzy Rule Base System at a glace Figure 12. Rule surface 4850
VI. RESULTS I this sectio we shall compare the proposed scheme with the well-kow adaptive techiques as well as oadaptive techiques. Power per subcarrier i assumed to be flat i the simulatios. Rayleigh flat fadig chael is used for simulatios. Fig-13 shows sigificace of the scheme over fixed modulatio ad codig. I this case, same modulatio code pair is used for all OFDM subcarriers after each adaptatio iterval. I this simulatio the value for fixed modulatio code pair (MCP) is foud by same fuzzy rule base system. I this sceario the decisio of ext MCP is take o behalf of average received SNR from all subcarriers ad the suggested MCP is used for all the subcarriers regardless of their idividual chael coditios. I this way some subcarriers may have the appropriate MCP but some may ot, sice the decisio is made o average chael coditios rather tha idividual chael coditios. Fig-14 shows the compariso of FRBS assisted adaptive codig ad modulatio with various well kow schemes. First compariso is doe with [14], where authors used Coded Bisectio Method to choose the appropriate modulatio code pair for ext trasmissio. Secod compariso is give with [5], i which there were used Switchig Thresholds to adaptive the appropriate modulatio scheme for ext iterval. I this scheme subcarriers were grouped, the based upo the average group chael coditios modulatio was chose for that group. Though i this paper there give coded modulatio as well that is adaptive modulatio (AM) with fixed code rate but we compare it with the ucoded case oly, that was the best case i that paper. Third compariso is give with aother adaptive modulatio scheme that was proposed by Krogold [15]. I this techique decisio for suitable modulatio was take by virtue of average received SNR the switchig threshold decided the type of modulatio. This result shows that oly Adaptive Modulatio could ot achieve that throughput aloe which ca be achieved by combied adaptive codig ad modulatio. A sigificat superiority of the proposed scheme ca be observed i the simulatio. Lastly, FRBS performace was compared with o-adaptive case, there is superiority of proposed scheme was more tha 1.5bits/s/Hz. Performace of Fuzzy Rule Base System for differet fixed Quality of Services for all subcarriers is demostrated i Fig-15. It shows that for high power per subcarrier ad relatively poor QoS values throughput of the system approaches above 5bits/s/Hz while for high QoS achievable throughput is 2.5bits/s/Hz at 30dB of power per subcarrier Fig-16 reveals the performace of FRBS for differet QoS per subcarrier, which is more practical sceario. I this case each subcarrier has a differet QoS demad. Results show that with eve a diverse sceario, performace of Fuzzy Rule Base System is remarkable ad more tha 4bits/s/Hz throughput is achievable. This very performace 2 is compared with two fixed QoS for all subcarriers that is 10 5 ad 10. These two cases put the lower ad upper 5 boud o its performace, respectively. It meas the performace of radom QoS caot be more tha 10 ad less 2 tha 10. This sceario is practical i s sese that it perfectly matches with IEEE wireless stadards like IEEE802.16 (fixed) [16] ad IEEE802.16/e (mobile) [17], where variety of mobile users havig differet receptio power levels ad differet Quality of Service Demads are preset. I ordiary adaptive schemes, such chaels are chose to shut dow where chael coditios are very poor ad resources like power are redistributed amog rest of the subcarriers [5] [13]. But as far as the proposed scheme is cocered trasmissio is eve possible at chael with such a hostile sceario. Figure 13. Compariso of FRBS based AMC with Fixed MC usig QoS=10e-2 ad N=256 4851
Atta-ur-Rahma et al., 2012 Figure 14. Compariso of FRBS based AMC with Coded Bisectio AMC, switchig threshold based AM, Krogold Adaptive Modulatio ad No adaptive scheme, where QoS=10e-2 ad N=1024 Figure 15. Performace of FRBS based AMC for differet fixed QoS for all the subcarrier ad N =256 Figure 16. Performace compariso of FRBS based AMC for differet QoS for each subcarrier with fixed QoS for all subcarriers while N =256 4852
VII. CONCLUSIONS Fuzzy Rule Based System assisted Adaptive Codig ad Modulatio scheme is proposed ad aalyzed i cotrast to other adaptive techiques as well as fixed techiques. It is show that FRBS is more powerful i terms of 2 3 4 throughput ad bit error rate especially whe bit error rates are10,10,10. Moreover, i literature there is o aalytical solutios exists that icorporate both codig gai ad spectral efficiecy i terms of power per subcarrier ad bit error rate due to its highly o-liear ature. Fuzzy system o the other had has specialty to outperform i such vague scearios. So cost fuctio is formulated ad optimized that maximize the system s throughput uder certai costraits. Due to its o-covex ature ordiary covex optimizatio techiques may ot solve it. Aother promiet feature of proposed scheme is that it best suits IEEE802.16/e (WiMAX) stadards where there are fixed as well as mobile users havig differet quality of service demads or where some users may be more/less privileged i terms of speed ad data rates. Fuzzy Rule Base System performs woderful i such a diverse sceario where each subcarrier has its ow chael state as well as Quality of Service demad. VIII. ACKNOWLEDGMENT This research work is sposored by Higher Educatio Commissio (HEC), of Pakista. REFERENCES [1] I. Kalet, The multitoe chael, IEEE Tra. Commu., vol. 37, pp.119 124, Feb. 1989. [2] P. S. Chow, J. M. Cioffi, ad J. A. C. Bigham, A practical discrete multitoe trasceiver loadig algorithm for data trasmissio over spectrally shaped chaels, IEEE Tras. Commu., vol. 48, pp. 772 775, 1995. [3] A.Cyzlwik, Adaptive OFDM for widebad radio chaels, Global Telecommuicatios Coferece, vol. 1, pp. 713-718, Nov 1996. [4] R.F.H. Fischer ad J.B. Huber, A New Loadig Algorithm for Discrete Multitoe Trasmissio, i IEEE Global Telecommuicatios Coferece, Lodo, Eglad, November 1996, pp. 724 728. [5] FAEZAH, J., SABIRA, K. (2009) Adaptive Modulatio for OFDM Systems. Iteratioal Joural of Commuicatio Networks ad Iformatio Security (IJCNIS) Vol. 1, No. 2. [6] T. H. Liew, C.H. Wog ad L. Hazo, Block turbo coded burst-by-burst adaptive modems. Proc. Microcoll 99, Budapest, Hugary, pp. 59 62, 1999. [7] T. Keller ad L. Hazo, Adaptive modulatio techiques for duplex OFDM trasmissio. IEEE Trasactios o Vehicular Techology, Vol.49, o.5, pp.1893-1906, 2000 [8] Y. Li ad W.E. Rya, Mutual-Iformatio-Based Adaptive Bit-Loadig Algorithms for LDPC-Coded OFDM, IEEE Tras. o Wireless Commuicatios, vol. 6, o. 5, pp. 1670-1680, May 2007 [9] G. Caire, G. Taricco, ad E. Biglieri, Bit-Iterleaved Coded Modulatio, IEEE Tras. o Iformatio Theory, vol. 44, o. 3, pp. 927-946, 1998. [10] C. Stierstorfer ad R.F.H. Fischer, (Gray) Mappig for Bit-Iterleaved Coded Modulatio, i IEEE Vehicular Techology Coferece (VTC2007-Sprig), Dubli, Irelad, April 2007. [11] S. Stiglmayr, M. Bossert, ad E. Costa, Adaptive Codig ad Modulatio i OFDM Systems usig BICM ad Rate-Compaitble Puctured Codes, i Europea Wireless, Paris, Frace, April 2007. [12] Y. Lei ad A. Burr, Adaptive Modulatio ad Code Rate for Turbo Coded OFDM Trasmissios. Vehicular Techology Coferece VTC2007 pp.2702-2706, 2007. [13] O. Al-Askary, Codig ad iterative decodig of cocateated multi-level codes for the Rayleigh fadig chael, i Doctoral thesis i Radio commuicatio systems, Stockholm, Swede: KTH Iformatio ad Commuicatio Techology, 2006. [14] C. Bockelma, D. W ubbe ad K. D. Kammeyer, Rate Ehacemet of BICM-OFDM with Adaptive Coded Modulatio via a Bisectio approach. IEEE 10th Workshop o Sigal Processig Advaces i Wireless Commuicatios, SPAWC '09. pp. 658 662, 2009. [15] B.S. Krogold, K. Ramchadra ad D.L. Joes, Computatioally Efficiet Optimal Power Allocatio Algorithm for Multicarrier Commuicatio Systems, IEEE Tras. o Commuicatios, vol. 48, o. 1, pp. 23 27, Jauary 2000. [16] IEEE Std 802.16TM-2004, Part 16: Air iterface for fixed broadbad wireless access systems, Oct 2004. [17] IEEE Std 802.16Etm-2005, Part 16: Air iterface for fixed ad mobile broadbad wireless access systems, Feb. 2006. 4853