Energy Effcent Adaptve Modulaton n Wreless Cogntve Rado Ad Hoc Networks Song Gao, Ljun Qan*, Dhadesugoor. R. Vaman ARO/ARL Center for Battlefeld Communcatons Research Prare Vew A&M Unversty, Texas A&M Unversty System Prare Vew, Texas 77446, USA Abstract In ths paper, an energy effcent adaptve modulaton scheme s proposed for a wreless cogntve rado ad hoc network, where each node s equpped wth cogntve rado and the network s an OFDMA system operatng on tme slots. In each slot, the users wth new traffc demand wll sense the spectrum and locate the avalable subcarrer set. Then they choose subcarrers wth favorable channel condton ndvdually whle avod ntroducng harmful nterference to the exstng users. Gven the delay requrement, an adaptve modulaton strategy s proposed for each ndvdual user to mnmze the energy consumpton per bt over the avalable subcarrer set by selectng the optmal constellaton sze. The optmal soluton of energy effcent adaptve modulaton s derved by usng Dnkelbacktype algorthm, and a sub-gradent search algorthm s proposed to locate the optmal constellaton sze. Furthermore, a suboptmal constant bt allocaton strategy s presented to address the delay constrants and a dstrbuted power control s performed to manage the co-channel nterference among new users when needed. Smulaton results demonstrate the effectveness of our approach. I. INTRODUCTION Msson Crtcal Networkng (MCN s under ntensve research recently due to ts wde-spread applcatons such as n mltary operatons, dsaster relef, etc. Usually MCN requres fast deployment as well as wthout nfrastructure support. Ths makes wreless ad hoc network a promsng canddate for MCN. Furthermore, effcent operaton of MCN plays a pvotal role because such networks are typcally resource (such as energy and spectrum constrant. In ths work, dstrbuted resource allocaton problem s consdered through jont desgn of adaptve modulaton and cogntve rado for an energy and spectrum constrant wreless ad hoc network to maxmze energy effcency and spectrum utlzaton. Cogntve Rado (CR [1] provdes the capablty of accessng the spectrum opportunstcally and greatly mproves spectrum utlzaton. Challenges arse wth such dynamc and herarchcal means of accessng the spectrum, especally for the dynamc resource allocaton of CR users by adaptng ther transmsson parameters to the varyng spectrum condton whle adherng to qualty of servce (QoS requrements [3]. In ths paper, an energy constraned wreless CR ad hoc network s consdered, where each node s equpped wth CR and has lmted battery energy. One of the crtcal performance *Correspondng author. Phone: 936-61-9915; Fax: 936-61-9930; E-mal: ljunqan@eee.org measures of such networks s network lfetme. Hence energy effcent resource management schemes are desred. In ths context, the present paper provdes a framework for energy effcent adaptve modulaton technques n wreless CR ad hoc networks that employ orthogonal frequency dvson multple access (OFDMA []. OFDMA s well suted for cogntve rado because t s agle n selectng and allocatng subcarrers dynamcally and facltates decodng at the recevng end of each subcarrer. The CR OFDMA network operates on tme slots. Exstng users transmt a plot sgnal perodcally on occuped subcarrers. By detectng the presence of such a plot sgnal, emergng CR users can determne the avalable subcarrer set n a target spectral range [4], and then tune transmsson parameters to map the nformaton bts effcently over the avalable subcarrer set. Adaptve modulaton, also called bt loadng, has been extensvely explored n prevous works. The basc dea behnd adaptve modulaton algorthm s to ensure that the most effcent mode s always employed over varyng channel condtons. Compared wth non-adaptve methods whch requre a fxed margn to mantan acceptable performance when the channel qualty s poor, adaptve approaches result n better effcency by takng advantage of the favorable channel condtons. In [5], [7], adaptve modulaton technques n MIMO- OFDM systems s presented wth the objectve to mnmze the transmsson power or mprove spectral effcency. The performance gan by combnng adaptve modulaton and power control s studed n [8], where sgnfcant throughput advantage has been demonstrated n the mult-user cellular system. Energy effcent modulaton optmzaton problem n sngle carrer system s nvestgated n [10] for uncoded and coded MQAM and MFSK. In [6], a varable-rate and varable-power MQAM modulaton scheme for hgh speed data transmsson s presented over fadng channels. In ths paper, adaptve modulaton s employed to optmally dstrbute bts to the transmtted symbols, hence determnng the constellaton sze on each of the avalable subcarrers. Adaptng the constellaton sze wll drectly nfluence the power consumpton of each node, and n turn wll affect the lfetme of the network. The optmalty s defned as maxmzng energy effcency subject to a pre-specfed bt error rate (BER and delay constrant. Specfcally, gven
requred BER and delay constrants, and assumng QAM s adopted as the modulaton scheme, the optmal modulaton order of QAM on each selected subcarrer that mnmze the energy for delvery of L bts per slot (thus maxmze the network lfetme s derved analytcally and s verfed by smulatons. The remander of ths paper s organzed as follows. In secton II, the system model and the problem formulaton are gven. A fully dstrbuted adaptve modulaton algorthm for each ndvdual user s proposed n secton III. In Secton IV, a dstrbuted power control algorthm s suggested to manage potental co-channel nterference caused by concurrent new users. Secton V contans the smulaton results and dscussons. Secton VI gves concludng remarks. II. SYSTEM MODEL We consder an energy constraned CR OFDMA network of N communcatng user-pars. Both transmtter and recever j s ndexed by N := {1,,..., N}. Ifj =, recever j s sad to be the ntended recever of transmtter. The transmsson system s assumed to be a tme-slotted OFDMA system wth fxed tme slot duraton T S. Slot synchronzaton s assumed to be acheved through a beaconng mechansm. Before each tme slot, a guard nterval s nserted to acheve synchronzaton, perform spectrum detecton as well as resource allocaton (based on the proposed scheme. Inter-carrer nterference (ICI caused by frequency offset from the sde lobes s not consdered n ths work (whch can be mtgated by wndowng the OFDM sgnal n the tme doman or adaptvely deactvatng adjacent subcarrers [11]. A frequency selectve Raylegh fadng channel s assumed at the physcal layer, and the entre spectrum s approprately dvded nto K subcarrers to guarantee each subcarrer experencng flat Raylegh fadng. We label the subcarrer set avalable to the transmtter recever par after spectrum detecton by L {1,,..., K} whch s a common subcarrer set avalable to both transmtter and ntended recever. Let G := { G k,j,,j N,k L } denote the subcarrer fadng coeffcent matrx, where G k,j stands for the sub-channel coeffcent gan from transmtter to recever j over subcarrer k. G k,j = Hk,j (f, where H,j k (f s the transfer functon. It s assumed that G adheres to a block fadng channel model whch remans nvarant over blocks (coherence tme slots of sze T S and uncorrelated across successve blocks. The nose s assumed to be addtve whte Gaussan nose (AWGN and to be ndependent of the symbols, wth varance σ for all recevers over the entre avalable subcarrer set. We defne p k to be the power allocated over subcarrer k for transmtter. The sgnal to nterference plus nose rato (SINR of recever over subcarrer k, γ k, can be expressed as γ k (p k =α k (p k j p k α k (p k G k, j = j,j N Gk j, pk j + (1 σ where α k s the channel state nformaton (CSI. α k can be measured at the recever sde and s assumed to be known by the correspondng transmtter through a recprocal common control channel. We assume the coherent recepton wth perfect carrer-phase estmaton and perfect fadng-value estmaton at the recever and the transmtter, thus the delays nvolved n obtanng the channel estmate at recever and of provdng the nformaton to the correspondng transmtter can be neglected. For MQAM, the number of bts per symbol s denoted as b = log M whch s defned over postve nteger values. A bound on the probablty of bt error rate for MQAM s gven by P b 4 ( 1 1 ( 3 Q b b b 1 εav, ( N 0 where P b s the bt error probablty (BER, ε av s the average energy per symbol, and Q(x = 1 x π exp( u du. Snce Q(x 1 x exp(, by approxmatng the bound as equalty we obtan P b = ( 1 1 ( 3 exp b b ( b 1 γ. (3 Thus, the transmsson power to guarantee a requred BER P,b on each selected subcarrers can be derved as ( ( bk 1 p k = 3 α k ln 1 1/ b k P,b bk. (4 In each tme slot, we assume L bts need to be transmtted by transmtter to acheve QoS requrements. On each subcarrer, the allocated bts s denoted as L k, thus k L L k = L. The number of MQAM symbols requred to send L k bts on subcarrer k s denoted as S k = Lk /bk. If the symbol perod s T ls, then S k = T,on /T,ls, T,on s the transmsson tme per slot and t s dentcal for all subcarrers. Furthermore, L k /b k = T,on /T,ls (5 If square pulses are used, T,ls can be approxmated by 1, B k where B k s the bandwdth of subcarrer k for transmtter. From (5, we obtan T,on = L k /(B k b k (6 In ths paper, wthout loss of generalty, we assume the subcarrers are equally dvded, and B k =1, k. Remark: From (5 and (6, the essental dea behnd the adaptve modulaton algorthm wth equally dvded bandwdth n OFDMA based CR networks s to allocate the same amount of symbols on each selected subcarrer whle wth dfferent constellaton sze accordng to the varyng subcarrer condtons.
Hence, the transmsson energy consumpton for transmtter over subcarrer k can be expressed as e k = p k T,on ( = T,on ( bk 1 ln 1 1/ = bk b k P,b ( ln 1 1/ bk b k P Lk (7,b 3α k ( bk 1 3α k In an energy constraned network (e.g. a wreless sensor network, recepton power s not neglgble snce t s generally comparable to the transmsson power [13]. In ths work, we denote the recevng power as p r whch s treated as a constant value for all recevers [1]. Thus, for transmtter and recever par, the total energy consumpton for the delvery of L bts s gven by e = k L e k + p r T,on = ( bk 1 3α k η(b k +p r T,on k L ( η(b k =ln 1 1/ bk b k P (8,b Based on (6 and B k =1, k, T,on can be further expressed as T,on = L k k L = L (B k b k b k k L k L Takng (9 nto (8, the energy consumpton for the delvery of L bts can be re-organzed as e = L ( bk 1 η(b k +p r (10 k L b k k L 3α k From the expresson of e, t s observed that the transmsson energy consumpton s ncreasng wth respect to b k,onthe contrary, the recepton energy s decreasng w.r.t. b k. Therefore, energy effcent bt loadng algorthm needs to locate an optmal trade-off between transmsson and recepton energy. In each tme slot, transmsson s requred to be bounded by the delay constrants,.e., T,on T S, whch leads to L b k (11 T S k L Defne L/T S := b mn. Gven the above system assumptons, we end up wth the followng constraned optmzaton b k (9 problem for transmtter recever par : mn b k Z+ e s.t. k L b k b mn, P b P,b, N (1 III. ENERGY EFFICIENT ADAPTIVE MODULATION SCHEME Snce the desgn varable b k s defned over postve ntegers, (1 s a constraned nteger fractonal programmng problem [9]. Due to the non-convex nature of the problem, hgh computatonal complexty s expected by exhaustve search of the optmal soluton, especally wth the ncrease of the number of avalable subcarrers. In ths paper, we propose an effcent one-dmensonal sub-gradent search algorthm by relaxng b k to postve real numbers R + whch wll provde the (best performance bound of energy effcency. For uncoded MQAM, f we approxmate η(b k by ( ln 1 1/ ( bk b k P 3 ln b b k P (13 b We can smplfy the representaton of total energy consumpton (10 as ( bk 1 e = L k L 3α k ( ln k L b k 3 b k P,b + p r (14 The relatve looseness caused by the bound (13 s less than 7% (see Fg.1 when b s wthn the range [, 10] (whch s a reasonable regon for practcal MQAM n wreless ˆb k networks. After we fnd the optmal soluton for the relaxed optmzaton problem, the optmal nteger result b k can be located by evaluatng the energy effcency at the two neghborng nteger ponts. Note that generally for nteger programmng problems, the optmal soluton may not be one of the neghborng nteger ponts. However, n the studed cases, the smulatons demonstrate the effectveness of the proposed relaxaton algorthm for fndng the true optmal soluton, and the obtaned optmal soluton n real doman can be treated as the performance bound. A. Unconstraned Energy Effcent Allocaton Algorthm In ths paper, we decouple the constraned fractonal programmng problem nto an unconstraned one and a branchand-bound algorthm s appled thereafter to gan nsghts of the optmal adaptve modulaton order. Based on the
to teratvely search for λ ( ˆb. From (19, t can be observed that orgnal J dmensonal fractonal programmng problem can be reduced to one dmensonal optmzaton problem f λ ( ˆb s consdered to be a constant value for each subcarrer k. Thus, we propose a one dmensonal effcent sub-gradent search algorthm to locate the optmal λ ( ˆb whch wll determne the optmal modulaton order vector ˆb n reverse. Specfcally, we propose the followng sub-gradent projecton teraton (ndexed by n for locatng λ ( ˆb ( f ( λ (n +1=λ (n+β(n ˆb (n g ( ˆb (n λ (n (0 Fg. 1. Looseness of bound approxmaton (equaton (13 Dnkelback-type algorthm [9], we defne λ ( ˆb := f (ˆb g (ˆb, λ ( ˆb := mn e b k (ˆb R+ f ( ˆb :=L ( k 1 3 3α k ln k L ˆb k P + p r,b g ( ˆb := k L ˆbk (15 whereˆs used to represent the varables n the unconstraned optmzaton doman, ˆb = [ˆb 1, ˆb,...,ˆb k ] denotes the unconstraned optmal modulaton orders over avalable subcarrer set for transmtter recever par. In the practcal regon of MQAM constellaton sze, t can be seen that f ( ˆb,g ( ˆb are postve, contnuous, and convex (the proof s gven n Appendx A. Gven the channel matrx G and nose varance σ, ˆb s defned as the optmal pont by satsfyng λ ( ˆb λ ( ˆb, ˆb R J+ (16 where J s the cardnalty of the avalable subcarrer set. To analyze the propertes of the fractonal programmng optmzaton, we ntroduce the functon ρ (λ, ˆb :R + R J+ R + whch s gven by ρ (λ, ˆb :=f ( ˆb λ ( ˆb g ( ˆb (17 Therefore, the optmal soluton of the unconstraned energy effcency optmzaton problem can by obtaned by dfferentatng ρ (λ, ˆb wth respect to ˆb k ρ (λ, ˆb = f ( ˆb λ ( ˆb (18 ˆb k ˆb k By settng ρ (λ, ˆb / ˆb k =0, we obtan ( (ˆb k ˆb k 1.5 1 3α k ln ln P,b ˆb = λ ( ˆb (19 k ˆbk Because computng the entre optmal constellaton sze ˆb s too expensve, a more effcent approach s proposed here where β(n s a postve step sze. Each λ encountered durng the search process must be evaluated to determne the constellaton sze assocated wth t whch can be used to update f ( ˆb (n and g ( ˆb (n, respectvely. From (0, the tentatve optmal modulaton order b k (n for the nth teraton can be derved over each subcarrer k L. And the temporary energy consumpton whch we denote as λ (n s updated wth the correspondng b k.theɛ-optmal constellaton sze b k has been reached f the followng condton holds for the subsequent {λ (n}. λ ( ˆb = λ(n = f ( ˆb (n g ( ˆb (n λ (n ɛ (1 where ɛ s an arbtrary small postve number. Smulatons demonstrate that the presented sub-gradent search algorthm provdes fast convergence and easy verfcaton when λ has been reached. The entre algorthm s summarzed as follows. Unconstraned Bt Loadng Algorthm 1 Intalzaton Intalze the unconstraned energy consumpton λ =0, the temporary energy consumpton λ =0, and the tentatve constellaton sze vector b =φ. Sub-Gradent search algorthm of λ Select λ (0 R + and n := 0 Take λ (n nto (19 for tentatve modulaton order allocaton b (n. Dervng the temporary energy consumpton λ (n by the correspondng b (n. Determne f λ (n and λ (n satsfy (1 for convergence If yes, stop and the unconstraned optmal constellaton sze over entre avalable subcarrer set s ˆb = b (n, and the optmal energy consumpton s λ ( ˆb = λ (n. If no, update λ (n based on (0, and go to the begnnng of step. Therefore, the optmal unconstraned constellaton szes set ˆb can be expressed as ˆb = max{ˆb k, 0}, k L, N ( k L
B. Constraned Energy Effcent Allocaton Algorthm The last secton provdes an effcent sub-gradent search algorthm for the unconstraned optmal constellaton sze ˆb =[ˆb 1, ˆb,...,ˆb k ], k L. Gven such unconstraned optmal soluton, the ultmate soluton space can be parttoned nto two sub-space based on the delay constrants. Case 1. If k L ˆbk b mn If the unconstraned soluton ˆb satsfes the delay constrants, t s regarded as the constraned optmal soluton by evaluatng the energy effcency at two neghborng nteger ponts when both of them meet the delay constrants. Otherwse, the constellaton sze n real doman wll be rounded up to the least greater nteger 1 to ensure the non-volaton of the delay constrants. arg mn{e ( ˆb k b k L ˆbk = ˆb k, k L, e ( ˆb k }, ˆb k b mn k L ˆb k k L <b mn Case. If k L ˆbk b mn When the unconstraned optmal soluton ˆb cannot satsfy the delay constrants, addtonal bts need to be allocated on the subcarrer set. Constraned optmzaton technques, such as the nteror-pont method [10], can be appled but wth consderable computatonal complexty. In order to acheve tractable complexty and mplementaton smplcty, we present a constant bt allocaton scheme based on the unconstraned optmal soluton whch wll allocate the extra requred bts equally over the entre selected subcarrer set. Due to the optmalty of the unconstraned soluton, mnmal ncreased bt allocaton whch s denoted as b wll result n the best energy effcency snce t has mnmal devaton from the optmal pont. Thus b s gven by b = b mn k L ˆbk b k = b Γ(ˆb, k L (3 where b k s the equally allocated bts on each subcarrer, and Γ(X s defned as the cardnalty of the nonzero elements n vector X. Eventually, the constraned modulaton order allocated on selected subcarrer can be expressed as ˆb = = ˆbk + b k, k L (4 k L b k k L IV. DISTRIBUTED POWER CONTROL ALGORITHM In secton III, an adaptve modulaton algorthm s proposed for each ndvdual user par. Whereas, n mult-user ad hoc networks, smultaneous spectrum access among multple transmtter recever pars n the same tme slot wll cause co-channel nterference and result n degradaton of network 1 A better soluton may be obtaned by searchng all the combnatons of b as long as k L b k b mn. However, the ncurred complexty s hgh. TABLE I CONVERGENCE OF THE PROPOSED SUB-GRADIENT SEARCH ALGORITHM Fg.. Iteraton No. b (n λ (n λ (n 1 [1.4, 1.5] 0.01 0.0345 [3.4, 3.5] 0.0345 0.063 3 [.9, 3.1] 0.063 0.055 4 [.9, 3.1] 0.055 0.055 Optmal soluton for energy per bt n two subcarrer case performance. In order to guarantee the QoS requrements, such as the delay constrants and BER, a dstrbuted power control algorthm s presented n ths secton to manage the potental co-channel nterference. Each emergng new user ntally obtan ther constellaton sze ndvdually wth the algorthm proposed n secton III, and then terates the dstrbuted power control algorthm to mtgate the nterference from peers. The power control algorthm s gven by [14] { } γ p k k (t + 1 = mn γ k(tpk (t,p max (5 where γ k s the ndvdual target SINR of transmtter over each subcarrer k whch can be determned by (3. Durng the power control stage, f the target SINR γ k can not be mantaned when transmtter hts ts power bound, the network s consdered to be unable to accommodate all the current new users. A mult-access control (MAC scheme s then requred to guarantee the farness among the users. p max V. SIMULATION RESULT In ths secton, we consder a wreless sensor network wth cogntve rado capablty. The channel gans are assumed to be sampled from a Raylegh dstrbuton wth mean equals to 0.4d 3, where d s the dstance from the transmtter to the recever. The entre spectrum s equally dvded nto subcarrers wth bandwdth 100 KHz for each subcarrer. The duraton of each tme slot T S s assumed to be 10ms n whch L bts need to be transmtted. The thermal nose power s assumed to be the same over all subcarrers and equal to 10 8 W.
TABLE III COMPARISON BETWEEN EXHAUSTIVE SEARCH RESULTS AND PROPOSED SUB-GRADIENT RESULTS No. of Sub. ˆb,r b,z b,opt 1 [3.56] [4] [4] [3.11,.9] [3, 3] [3, 3] 3 [.75,.61,.5] [3,3,] [3,3,] 4 [.5,.43,.3,.1] [3,,,] [3,,,] 5 [.31,.7,.,.1, ] [,,,,] [,,,,] We frst nvestgate the fast convergence and computaton effcency of the proposed algorthm. Wthout loss of generalty, we assume the avalable subcarrer set after spectrum sensng ncludes two subcarrers experencng Ralyegh fadng. By adoptng the step sze β(n =1of (0, the smulaton (Table I shows the convergence occurs n 4 steps. And through Fg., the mnmal energy effcency s acheved at the optmal constellaton sze vector n two-subcarrer case. Even though the subcarrer set s small n ths smulaton, comparable convergence speed s expected wth the expanson of the avalable subcarer set as shown n (Table II for a 10- subcarrer case. In Fg. 3 and Table III, we compared the performance between the optmal soluton obtaned by exhaustve search and the soluton obtaned by the proposed sub-gradent search algorthm. The system wth the avalable subcarrer set rangng from 1 to 5 subcarrers s shown n ths llustraton. We assume b mn =4. The dashed lne marked wth crcle represents the optmal energy effcency acheved when b s relaxed to real doman and t acts as the performance bound for the achevable energy effcency. The sold lne marked wth square s the acheved energy effcency when b s n the nteger doman. It can be observed that wth the ncrease of the number of subcarrers, the energy effcency s mproved from 3.84 10 7 to 1.88 10 7 at the expense of avalable bandwdth. In Table III, b,opt s the optmal soluton obtaned from exhaustve search that serves as a benchmark. It s observed that the optmal soluton derved by relaxaton (b,z sthe same as that n exhaustve search (b,opt, whch demonstrates the effectveness of the proposed algorthm. Furthermore, the performance gap between the performance bound obtaned by ˆb,r and the acheved energy effcency by usng b s very small as shown n Fg. 3. In Table IV, the constraned case (Secton III-B s demonstrated wth the same system settng as the unconstraned case (Table III except for b mn =10. After obtanng the unconstraned optmal soluton followng the proposed subgradent search algorthm, the unfulflled bts s equally loaded on the avalable subcarrer set whch provdes a suboptmal soluton for the bt loadng wth tractable complexty. It can be observed that even the sub-optmal constant bt loadng strategy offers close performance to the optmal soluton acheved through exhaustve search. Another thng to be noted s that wth the ncrease of avalable subcarrers, the constraned problem evolves nto an unconstraned one wth the ncrease of avalable bandwdth. In Table IV, the fst four cases are Fg. 3. Performance bound vs. acheved energy effcency for b TABLE IV COMPARISON BETWEEN CONSTRAINED SUB-GRADIENT RESULTS AND EXHAUSTIVE SEARCH RESULTS No. of Sub. b,r b,z b,opt 1 [10] [10] [10] [4.98, 5.0] [5, 6] [5, 5] 3 [3.3733, 3.3533, 3.733] [4, 4, 4] [4,3,3] 4 [.595,.575,.505,.35] [3,3,3,3] [3,3,,] 5 [.7,.3,.,.16, 1.99] [,,,,] [,,,,] correspondng to the constraned problem Case, and the last case s n Case 1. In multple user scenaro, n order to nvestgate the dstrbuted power control for managng the co-channel nterference, we consder two emergng new users who share two common subcarrers as shown n Fg.4. After each user conductng the adaptve modulaton algorthm ndependently, dstrbuted power control s performed to guarantee the BER (P,b requrment by ncreasng the transmsson power. It can be observed that the system converges n 3-4 steps. Fg. 4. Convergence of dstrbuted power control
TABLE II CONVERGENCE OF THE PROPOSED SUB-GRADIENT SEARCH ALGORITHM Iteraton No. b (n λ (n λ (n 1 [.5,.8.3,.3,.48,.481,.5,.53,.58,.69] 0.005 0.015 [1.87, 1.91 1.9, 1.93,.08,.09,.14,.15,.18,.3] 0.015 0.0098 3 [1.8, 1.86 1.87, 1.88,.04,.05,.11,.1,.14,.6] 0.0098 0.0096 4 [1.8, 1.86 1.87, 1.88,.04,.05,.11,.1,.14,.6] 0.0096 0.0096 VI. CONCLUSION In ths paper, a fully dstrbuted energy effcent adaptve modulaton algorthm s proposed for a wreless OFDMA cogntve rado ad hoc network. A Dnkelback-type algorthm s proposed to solve the unconstraned optmzaton problem such that energy consumpton for data delvery (n terms of energy per bt s mnmzed over the avalable subcarrer set. A sub-gradent search algorthm s proposed to locate the optmal constellaton sze for each ndvdual user. Gven the delay requrement, a sub-optmal constant bt allocaton strategy s presented to resolve the constraned optmzaton problem. In addton, a dstrbuted power control s performed to manage the co-channel nterference among the new users when needed. Although the proposed algorthm provdes a sub-optmal soluton to the constraned optmzaton problem, t has low computatonal complexty and mplementaton smplcty. Furthermore, for the unconstraned cases, the proposed sub-gradent algorthm turns a mult-dmensonal optmzaton problem nto one-dmensonal search problem wth fast convergence speed. When the network s unable to accommodate the traffc load of all new users n the current tme slot, a schedulng algorthm wth farness consderaton s desred, whch wll be one of our future efforts. In addton, large scale smulaton experments wll be carred out as well. ACKNOWLEDGMENT Ths research work s supported n part by the Natonal Scence Foundaton under award 0531507 and by the U.S. Army Research Offce under Cooperatve Agreement W911NF-04- -0054. VII. APPENDIX A The convexty of the functon (over b. From (15, t s apparent g ( ˆb can be regarded as convex functon. For f ( ˆb, whch s expressed as f ( ˆb :=L k L (ˆb k 1 3α k For each subcarrer k, we defne f k (bk as ( 3 ln ˆb k P + p r,b ( f k (b k :=L ( ˆb k 1 3 ln ˆb k P + p r,b 3α k Fg. 5. convexty of functon The second dervatve of f k(bk over practcal regon [, 10] s gven as ( 3α k L f k(bk ( 3 b k = bk ln ln ˆb k P,b bk +1 ln b k + bk 1 ( b k (6 It s shown n Fg.5, that f k (bk > 0 for [, 10]. Therefore, ( b k f k(bk s convex functon whch wll result n the convexty of functon f ( ˆb. REFERENCES [1] J. Mtola et al, Cogntve rado: makng software rados more personal, IEEE Personal Communcatons, Vol. 6, Issue 4, pp. 13-18, Aug. 1999. [] L. Ltwn, and M. Pugel, The Prncples of OFDM, RF Sgnal Processng, pp. 30-48, Jan. 001. [3] S. T. Chung and J. M. Coff, Rate and Power Control n a Two- User Multcarrer Channel Wth No Coordnaton: The Optmal Scheme Versus a Suboptmal Method, IEEE Trans. on Communcatons, Vol. 51, No. 11, pp. 1768-177, Nov. 003. [4] S. Gao, L. Qan, D. R. Vaman and Q. Qu, Energy Effcent Adaptve Modulaton n Wreless Cogntve Rado Sensor Networks, IEEE Internatonal Conference on Communcaton, pp. 3980-3986, June 007. [5] A. Pandharpande, Adaptve modulaton for MIMO-OFDM systems, IEEE Vehcular Technology Conference, Vol., Issue 6-9, pp. 166-170, 004. [6] A. J. Goldsmth, S. Chua, Varable-Rate Varable-Power MQAM for Fadng Channels, IEEE Transactons on Communcatons, Vol. 45, No. 10, pp.118-130, Oct. 1997. [7] S. Catreux, V. Erceg, D. Gesbert and R. W. Heath Jr., Adaptve modulaton and MIMO codng for broadband wreless data networks, IEEE Communcaton Magazne, pp.108-115, June 00.
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