Distributed Energy Efficient Spectrum Access in Cognitive Radio Wireless Ad Hoc Networks

Transcription

1 Dstrbuted Energy Effcent Spectrum Access n Cogntve Rado Wreless 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, the resource allocaton problem s consdered for energy effcent spectrum access n an energy constraned wreless cogntve rado ad hoc network, where each node s equpped wth cogntve rado, has lmted battery energy, 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. Gven the data rate requrement and maxmal power lmt, a constraned optmzaton problem s formulated for each ndvdual user to mnmze the energy consumpton per nformaton bt over all selected subcarrers, whle avod ntroducng harmful nterference to the exstng users. Because of the mult-dmensonal and non-convex nature of the resource allocaton problem, a fully dstrbuted subcarrer selecton and power allocaton algorthm s proposed by combnng an unconstraned optmzaton procedure wth a branch-and-bound method that parttonng the soluton space accordng to the power and rate constrants n order to obtan the constraned optmal soluton. Due to the noncooperatve behavor among new users, they wll execute dstrbuted power control to manage the cochannel nterference when needed. Smulaton results demonstrate that the proposed scheme performs tghtly to the centralzed optmal soluton n a mult-user envronment. In addton, the comparson between the proposed energy effcent allocaton scheme and the well establshed rate or power effcent allocaton algorthms s carred out to demonstrate the advantage of the proposed scheme n terms of network lfetme. Index Terms resource allocaton, cogntve rado, ad hoc network, OFDMA. An earler verson of the manuscrpt was publshed n IEEE WCNC Aprl 13, 2008

2 1 I. INTRODUCTION Although the U.S. government frequency allocaton data [1] shows that there s ferce competton for the use of spectra, especally n the bands from 0 to 3 GHz, t s ponted out n several recent measurement reports that the assgned spectrum are hghly under-utlzed [2], [3]. The dscrepancy between spectrum allocaton and spectrum use suggests that spectrum access s a more sgnfcant problem than physcal scarcty of spectrum, n large part due to legacy command-and-control regulaton that lmts the ablty of potental spectrum users to obtan such access [2]. In order to acheve much better spectrum utlzaton and vable frequency plannng, Cogntve Rados (CRs are under development to dynamcally capture the unoccuped spectrum [5], [4]. Furthermore, n many challengng stuatons, the spectrum condton and usage nformaton may not be avalable a pror, such as n battlefeld applcatons [38]. It s up to the CR users to sense the spectrum and obtan the spectrum occupancy. Many 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 and recepton parameters to the varyng spectrum condton whle adherng to power constrants and dverse qualty of servce (QoS requrements (see, for example, [12], [13], [14], [15], [18]. 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 measures of such networks s the network lfetme. Moreover, due to the nfrastructureless nature of ad hoc networks, dstrbuted resource management scheme s desred to coordnate and mantan communcatons between each transmttng recevng par. In ths context, the present paper provdes a framework of dstrbuted energy effcent spectrum access and resource allocaton n wreless CR ad hoc networks that employ orthogonal frequency dvson multple access (OFDMA [34], [8], [9] at the physcal layer. OFDMA s well suted for CR because t s agle n selectng and allocatng subcarrers dynamcally and t facltates decodng at the recevng end of each subcarrer [26]. In addton, mult-carrer sensng can be exploted to reduce sensng tme [6]. The CR OFDMA network operates on tme slots. An exstng user transmts a plot sgnal perodcally on occuped subcarrers [27]. By detectng the presence of such a plot sgnal, emergng CR users can determne the avalable subcarrer set n a target spectral range, and then select subcarrers and transmsson parameters (based on the proposed algorthm wthout ntroducng harmful nterference to the exstng users [5], [7]. In ths work, the prmary users are users wth on-gong communcatons, whle the secondary users refer to CR users wth new traffc demand. After the emergng CR users start ther Aprl 13, 2008

3 2 data transmssons, they become prmary users [38]. Each emergng CR user wll select ts subcarrers and determne ts transmsson parameters ndvdually by solvng an optmzaton problem. The optmzaton objectve s to mnmze ts energy consumpton per bt 1 whle satsfyng ts QoS requrements and power lmts. Compared wth the power mnmzaton wth respect to target data rate constrants [12] or throughput maxmzaton under power upper bound [13], ths objectve functon, whch measures the total energy consumed for relable nformaton bts transmtted, s partcularly sutable for energy constraned networks where the network lfetme s a crtcal metrc. The mult-dmensonal and non-convex nature of the optmzaton problem n mult-carrer systems makes t more challengng than the throughput maxmzaton/power mnmzaton problems or the energy effcency problem n a sngle carrer system [21]. Hence, we propose a two-step algorthm by frst decouplng t nto an unconstraned optmzaton problem, and a branch and bound method s appled thereafter to obtan the constraned optmal soluton by parttonng the soluton space accordng to power and QoS constrants. Although the emergng CR users wll not cause harmful nterference to the exstng users, they may choose the same subcarrers n the same tme slot ndependently, and thus co-channel nterference may be ntroduced. In ths work, we allow multple new users to share the same subcarrers as long as ther respectve Sgnal-to-Interference-and-Nose-Rato (SINR s acceptable. Ths may be acheved by dstrbuted power control [17], whch converges very fast. The flow chart of the proposed dstrbuted energy effcent spectrum access and resource allocaton scheme s hghlghted n Fg. 1, where step 2 corresponds to the constraned optmzaton performed by each emergng user ndvdually. More detaled llustratons of the flow chart are gven n secton IV. The remander of ths paper s organzed as follows. In secton II, the system model and the problem formulaton are gven. A fully dstrbuted channel selecton and power allocaton scheme for sngle user case 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 smulaton results and dscussons. Related works are dscussed n secton VI. Secton VII gves concludng remarks. II. SYSTEM MODEL We consder an energy constraned CR OFDMA network of N communcatng pars. Both transmtter and recever j s ndexed by N := {1, 2,..., N}. If j =, recever j s sad to be the ntended recever 1 whch s defned as the rato of the total transmsson and recepton power consumpton over avalable subcarrer set to ts acheved throughput Aprl 13, 2008

4 3 Fg. 1. Block dagram of the proposed dstrbuted resource allocaton algorthm 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 of the sde lobes pertanng to transmtter s not consdered n ths work (whch can be mtgated by wndowng the OFDM sgnal n the tme doman or adaptvely deactvatng adjacent subcarrers [19]. A frequency selectve Raylegh fadng channel s assumed at the physcal layer, and the entre spectrum s approprately dvded nto M subcarrers to guarantee each subcarrer experencng flat Raylegh fadng [10]. We label the subcarrer set avalable to the transmtter recever par after spectrum detecton { } by L {1, 2,..., M}. 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 2, where H,j k (f s the transfer functon [33]. It s assumed that G s avalable to Aprl 13, 2008

5 4 a central agent and 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, wth varance σ 2,k over subcarrer k of recever. We defne P := { p k, pk 0, N, k L } as the transmsson power allocaton matrx for all users n N over the entre avalable subcarrer set N L, where p k. For each transmtter, the power vector can be formed as If the k th subcarrer s not avalable for transmtter, p k s the power allocated over subcarrer k for transmtter p := [p 1, p 2,..., p M ] T (1 = 0. Each node s not only energy lmted but also has peak power constrant,.e., k L p k pmax. The set of all feasble power vector of transmtter s denoted by P P := p [0, p max ], p k p max, p k 0 k L k L The sgnal to nterference plus nose rato (SINR of recever over subcarrer k (γ k can be expressed as (2 where α k nose. α k γ k (p k = α k (p k j p k α k (p k j = G k, j,j N Gk j, pk j + σ2,k s defned as the channel state nformaton (CSI whch treats all nterference as background can be measured at the recever sde and s assumed to be known by the correspondng transmtter through a recprocal common control channel. When all users dvde the spectrum n the same fashon wthout coordnaton, t s referred to as a Parallel Gaussan Interference Channel [20] whch leads to a tractable nner bound to the capacty regon of the nterference model. The achevable maxmum data rate for each user (Shannon s capacty formula s c (p B k = c k ( p k = log 2 (1 + α k (p k j p k k L B k k L, p k P where B k s the equally dvded subcarrer bandwdth for transmtter. Wthout loss of generalty, B k s assumed to be unty n ths work. The nose s assumed to be ndependent of the symbols and has varance σ 2 for all recevers over entre avalable subcarrer set. Furthermore, all communcatng transmtter and recever pars are assumed to have dverse QoS requrements specfed by k L c k rtar, where r tar s the target data rate of transmtter. (3 (4 Aprl 13, 2008

6 5 In an energy constraned network (such as a wreless sensor network, recepton power s not neglgble snce t s generally comparable to the transmsson power [22], [21], [23]. In ths work, we denote the recevng power as p r whch s treated as a constant value for all recevers [21]. Amng at achevng hgh energy effcency, the energy consumpton per nformaton bt for transmtter recever par n each tme slot s k L e (p, c := p k + pr k L c k (5 Let S (p, c denote the set of all power and rate allocatons satsfyng QoS requrements and power lmt constrants for transmtter, and t s gven by S (p, c = { p, c : p P, c r tar, N } (6 Gven the above system assumptons and the objectve defned n (5, we end up wth the followng constraned optmzaton problem. mn e (p, c p k,ck S s.t. c (p r tar, N p k p max, N (7 k L III. OPTIMAL SUBCARRIER SELECTION AND POWER ALLOCATION The problem (7 s a combnatoral optmzaton problem and the objectve functon s not convex/concave. Constraned optmzaton technques, such as mult-dmensonal nteror-pont method [32], can be appled here but wth consderable computatonal complexty. Hence, we propose a two-stage algorthm to decouple the orgnal problem nto an unconstraned problem n order to reduce the search space. After the optmal soluton for the unconstraned problem s obtaned n stage 1, the power and data rate constrants wll be examned n search of the fnal optmal soluton. It should be noted that the soluton of the unconstraned problem provdes the optmal operatng pont whch can be taken as the benchmark for the system desgn. A. Unconstraned Energy Effcent Allocaton We defne the unconstraned energy per bt functon as ˆp k + p r k L f(ˆp, α := ( (8 log α k ˆp k k L Aprl 13, 2008

7 6 whereˆs used to represent the varables n the unconstraned optmzaton doman and α = k L α k. It s assumed f ( ˆp, α s a contnuous functon n R + M. We defne the unconstraned optmal energy per bt for transmtter of (8 as ˆζ = mn f( ˆp, α. [ˆp 1 1 Energy Effcent Waterfllng: Theorem 1: Gven the channel state nformaton α and nose power, power allocaton ˆp =, ˆp 2,...,, k L ] s defned as the unconstraned optmal power allocaton by satsfyng f (ˆp, α f (ˆp, α, ˆp R M+ (9 Then the unconstraned optmal power allocaton can be obtaned by solvng the followng equatons: { = max log e 2 ˆζ 1 } α k, 0 + p r ˆζ k L = ( (10 log α k k L Proof: Dfferentatng f( ˆp, α wth respect to ˆp k (whch stands for the power allocated for transmtter on subcarrer k, we obtan the equatons (10. The detals of the dervaton are gven n Appendx A. The value of ˆζ observed that ˆp can be obtaned by usng a numercal method whch wll n turn determne ˆp. It s has smlar type of rate-adaptve / margn-adaptve waterfllng results, and we name t energy-effcent waterfllng. Whereas, the fundamental dfference among them les n the postons of ther respectve optmal ponts. The rate-adaptve waterfllng maxmzes the achevable data rate under power upper bound, and margn-adaptve waterfllng mnmzes the total transmsson power subject to a fxed rate constrant [16], both of whch acheve ther optmalty at the boundary of the constrants. On the contrary, the proposed energy-effcent waterfllng selects the most energy-effcent operatng pont (n other words, t selects the optmal data rate that mnmzes the energy consumpton per nformaton bt whle adherng to the QoS requrements and power lmts. In ths case, optmalty s usually obtaned n the constrant nterval rather than on the boundary. In fact, the rate-adaptve and margn-adaptve waterfllng can be consdered as specal cases of the energy-effcent waterfllng solved n ths paper. If we set k L p k = p con p max or k L c k (pk = rtar, the energy-effcent allocaton problem s reduced to the well explored rate-adaptve or margn-adaptve waterfllng problem. 2 Feasblty Regon: The exstence of the soluton for the unconstraned optmzaton (mn f(ˆp, α depends on the subcarrer condton α k f we assume other system parameters (e.g. bandwdth, maxmal Aprl 13, 2008

8 7 power, etc. are fxed. From (10, f we take ˆp nto the expresson of ˆζ, we can get ˆζ = I( = Γ(ˆp log e 2 ˆζ k L 1 α k I( + p r Γ(ˆp log 2 (log e 2 ˆζ + k L log 2 (α k I( 1, > 0 0, 0 (11 where Γ(X s defned as the cardnalty of nonzero elements n vector X. If we defne g(ˆζ as g(ˆζ = Γ(ˆp log 2 (log e 2 ˆζ + log 2 (α k I( ˆζ Γ(ˆp log e 2 ˆζ k L k L 1 α k I( + p r, the optmal soluton ˆζ can be determned by settng g(ˆζ = 0, and the exstence of the optmal soluton s nfluenced by the subcarrer condton α k. Ths s llustrated n Fg.2. A unque optmal soluton (ˆζ,2 s obtaned when the subcarrer condton s good; whle no feasble soluton exsts when the subcarrer condton s bad. Multple solutons may be obtaned when the subcarrer condton s n the mddle range. In ths case, only the larger soluton (ˆζ,3 s the feasble soluton, and ths can be verfed by checkng the correspondng power allocaton,.e., all the allocated power should be non-negatve. The feasblty condton of the unconstraned optmzaton problem s gven n the followng theorem. Theorem 2: Denote the maxmal optmal soluton of ˆζ subcarrer as α τ, ατ as ˆζ max (12 and the channel gan of the best = max{α k, k L }. The feasblty condton for the exstence of the optmal soluton of the energy effcent waterfllng (10 s gven by α τ ln 2. Proof: 1 Necessty: From the optmal soluton of energy effcent waterfllng (10, t s observed the amount of allocated power s determned by the subcarrer condton α k, specfcally, more power should be allocated on better subcarrer. Thus, f the optmal soluton exsts, at least the power allocated ˆζ max on the best subcarrer should be non-negatve,.e., ˆp τ = loge 2 ˆζ max 1 α τ 0 = α τ ln 2. 2 Suffcency: We prove ths part by contradcton. If α τ ˆζ ln 2 and stll no optmal soluton exsts, max whch mples that the power allocated on the entre subcarrer set s negatve,.e., < 0, k L, then ˆζ max 1 α < 0 = α τ τ < ln 2, whch contradcts the condton α ˆζ τ max ˆζ ln 2. Ths completes the max proof. Theorem 2 suggests that t s suffcent to check the best avalable subcarrer n order to determne the feasblty of the unconstraned optmzaton problem. ˆζ max Aprl 13, 2008

9 8 Fg. 2. Feasble soluton vs. subcarrer condton B. Constraned Energy-Effcent Allocaton Algorthm Gven the unconstraned optmal soluton ˆp, N, the prevous secton offers the optmal operatng pont wth best energy effcency of each ndvdual user. However, some users may not satsfy ther respectve data rate and/or power constrants when operatng at ths pont. In ths secton, we partton the soluton space of the constraned optmzaton problem (7 nto four sub-spaces based on the power and data rate constrants, as hghlghted n Fg.3. 1 p max and c k ( r tar. k L k L In ths case, the unconstraned optmal soluton ˆp 2 3 of (10 satsfes the sum-power and rate requrement constrants. Apparently ˆp s the optmal soluton of the orgnal problem (7. p max and c k ( r tar. k L k L In ths case, the allocated power has already exceeded the sum-power constrant, but the rate requrement s stll not met, even under the optmal subcarrer selecton and power allocaton. Therefore, there s no feasble soluton for the orgnal problem (7. p max and c k ( r tar. k L k L If both the power allocated on all subcarrers does not reach the maxmal power bound and the Aprl 13, 2008

10 9 Fg. 3. Partton of the soluton space of the constraned optmzaton problem data rate requrement s not met, the power should be ncreased to acheve data rate requrement under the maxmal power bound. Based on (7 and (10, we can modfy the orgnal problem as ( + p k + p r mn ˆp k + pk S k K k K log 2 s.t. k K c k ( k K ( 1 + α k ( + p k + p k r tar, N ( + p k p max, N (13 where K s defned as the selected subcarrer set through the optmal energy effcent waterfllng soluton, K L. If we ncrease the power on any one of the subcarrers, such as the k th subcarrer, the correspondng constraned energy consumpton per bt can be expressed as p k + k K + p r ζ k = ( log α k + c k k K ( 1 + α c k k = log (ˆp k + p k α k ˆpk (14 Aprl 13, 2008

11 10 From (10, c k can be smplfed to c k = log 2 (1 + pk. It s observed that gven the log e ˆζ 2 ncreased power p k on subcarrer k, the ncreased data rate does not rely on ts subcarrer condton α k, snce loge 2 ˆζ s a constant value for the entre selected subcarrer set. In other words, for any two subcarrer k, l K of transmtter N, f p k = p l, then ck = c l. And the constraned energy consumpton per bt ζ k and ζ l wll not vary due to dfferent chosen subcarrers. If we presume, n order to reach the data rate requrement r tar, the addtonal requred power p over the selected subcarrer set s known and denoted as p = k K p k. Then, problem (13 s equvalent to mn ˆp k + pk S k K log 2 s.t. k K c k ( k K p + k K ( 1 + α k + p r + c k k K + p k r tar, N ( + p k p max, N (15 If we assume p has been pre-determned, n order to mnmze energy consumpton per bt ζ, k K c(ˆp + k K c k need to be maxmzed. In other words, maxmzng k K c k wll result n a classcal rate-adaptve waterfllng problem. max log 2 (1 + pk k K log e 2 ˆζ s.t. ( + p k p max, N (16 k K Because log e 2 ˆζ s a constant value for the entre selected subcarrer set K, the soluton of the above water fllng problem mples that the optmal soluton p k for (16 should be the same for all chosen subcarrers. In other words, gven the total requred addtonal power p, the power should be equally allocated on all subcarrers, p k = p Γ(L. Thus, problem (13 can be rewrtten as mn k K log 2 s.t. k K p + k K ( 1 + α k + p r + c k k K + p p max, N (17 Aprl 13, 2008

12 11 Fg. 4. Illustraton of constraned ζ k wth respect to p where c k = Γ(K log 2 (1 + p Γ(K log e 2 ˆζ. Gven the unconstraned optmal soluton ˆp from stage 1, (17 can be consdered as an objectve functon n terms of varable p bounded by p max k K. Lemma 1: The constraned energy consumpton per bt of problem (17 whch s denoted as ζ s always worse than the unconstraned optmal energy effcency ˆζ wth respect to the power ncrease p k,.e. ζ ˆζ, p R +. The proof of Lemma 1 s gven n Appendx B. Due to the optmalty of the unconstraned soluton ˆp, the mnmal devaton from ˆζ wll result n the optmal energy effcency. Thus, the optmal power ncrease to satsfy the target data rate wll be the mnmal requred addtonal power as llustrated n Fg.4. Therefore, the optmal requred addtonal power (mn p to satsfy the data rate requrement r tar addtonal power p mn can be calculated as mn p = k K p k. The mnmal requred = mn p can be derved by p mn log 2 (1 + Γ(K log e 2 ˆζ = r tar k K c k ( Γ(K (18 Aprl 13, 2008

13 12 From (18, the optmal power ncrease on kth subcarrer p k s gven by p k log e 2 ˆζ = exp ( r tar k K c k (ˆpk log e 2 Γ(K 1 (19 If p mn exceeds the remanng power,.e., k K + p mn soluton for (7. If k K s + p mn p max, there s no feasble p max, the optmal soluton for the orgnal problem (7 p k = + p k (20 4 p max and c k ( r tar. k L k L In ths case, the data rate requrement s satsfed but the allocated power exceeds the lmt. In order to obtan a feasble soluton, the allocated power should be decreased. The dervaton of the optmal soluton follows smlar procedures as gven n case 3. We can re-organze the problem as mn ˆp k pk S k K k K log 2 s.t. k K c k ( k K ( p k + p r ( 1 + α k ( p k p k r tar, N ( p k p max, N (21 If we decrease power on one of the subcarrers, such as the k th subcarrer, we obtan k K ζ k p k = ( log α k + c k k K ( 1 + α c k k = log (ˆp k p k α k ˆpk (22 From (10, c k can be smplfed to c k = log 2 (1 p k / loge 2 ˆζ. Smlar to case 3, the decrease of the data rate does not depend on partcular subcarrer condton. In other words, for any two subcarrers k, l K of transmtter N, f p k = p l, then ck = c l. Hence, Aprl 13, 2008

14 13 problem (21 s smplfed to mn ˆp k pk S p + p r k K ( log α k + c k k K k K s.t. k K c k ( k K p k r tar, N ( p k p max, N (23 where p = k K p k. If we presume p s fxed, to mnmze energy consumpton per bt n (23, t s equvalent to maxmze c k = log 2 (1 p k / loge 2 ˆζ. We get a dual classcal water fllng problem. Therefore, the decreased power p should be equally reduced for the entre subcarrer set K. p k equvalent to = p /Γ(K, dm R (K = Γ(K for each subcarrer. Problem (21 s p + p r k K mn ( log α k + c k k K k K s.t. k K where k K c k = Γ(K log 2 (1 p p max, N (24 p Γ(K log e 2 ˆζ. The devaton from the unconstraned optmal pont ˆp wll lead to energy effcency degradaton (the proof s gven n Appendx C. Lkewse, the mnmal devaton from the unconstraned optmal pont ˆζ wll lead to the optmal soluton. Hence, the mnmal decreased power tll the maxmal power bound p max wll provde the optmal energy effcency for user as llustrated n Fg.5. When the power reaches the lmt p max, f the acheved data rate s stll below the target data rate r tar, the problem (21 s nfeasble. Otherwse, the optmal soluton s p s mn p = Γ(K, and the optmal decreased power p k on each subcarrer p k The optmal soluton for the orgnal problem (7 s p k = mn p k K = p k pmax (25 Γ(K Γ(K = p k. The nter-relatonshp and evolvement of the four cases parttoned by the power and data rate constrants are hghlghted n Fg.3. Excellent and terrble subcarrer condtons wll lead to case 1 (feasble and case 2 (nfeasble, respectvely. When the subcarrer condtons are good, the sold lnes from case 3 and case 4 lead the problem nto the feasble regon (case 1 of the constraned optmzaton Aprl 13, 2008

15 14 Fg. 5. Illustraton of constraned ζ wth respect to p problem when t reaches the maxmal power and target data rate bounds, respectvely. Whereas, the dashed lnes suggest that the problem enters the nfeasble regon (case 2 when the current subcarrer condton cannot accommodate the target data rate under the maxmal power lmts. IV. DISTRIBUTED POWER CONTROL In the prevous secton, each emergng new user obtans ts optmal subcarrer selecton and power allocaton ndvdually wthout consderng other new users. Although no nterference wll be ntroduced to the exstng users, due to the non-cooperatve behavor of each user, multple new users may choose the same subcarrers and co-channel nterference wll be ntroduced among themself. In order to mantan user s QoS, we propose an teratve and dstrbuted algorthm for reachng an equlbrum pont among multple transmtter and recever pars based on the dstrbuted power control scheme [17]. The dstrbuted power control algorthm s gven by p k (t + 1 = mn { } γ k γ k(tpk (t, p max (26 where γ k s the ndvdual target SINR of the th transmtter recever par over each subcarrer k, whch s determned by the constraned optmal soluton p, γ k = exp(ln 2 c(p k 1. Aprl 13, 2008

16 15 In the power control stage, each node only needs to know ts own receved SINR (γ k at ts desgnated recever to update ts transmsson power. Ths s avalable by feedback from the recevng node through a control channel. As a result, the proposed scheme s fully dstrbuted. Convergence propertes of ths type of algorthms were studed by Yates [17]. An nterference functon I(P s standard f t satsfes three condtons: postvty, monotoncty and scalablty. It s proved by Yates [17] that the standard teratve algorthm P (t+1 = I(P (t wll converge to a unque equlbrum that corresponds to the mnmum use of power. The dstrbuted power control scheme (26 s a specal case of the standard teratve algorthm. In summary, the proposed energy effcent spectrum access and resource allocaton scheme ncludes the followng steps, as hghlghted before n Fg. 1. Dstrbuted Energy Effcent Spectrum Access and Resource Allocaton 1 Intalzaton Each transmtter recever par obtans ther respectve avalable subcarrer set L through spectrum detecton. 2 Indvdual Energy Effcent Resource Allocaton Each transmtter recever par derves ts respectve unconstraned optmal soluton from equaton (10. Based on the power lmt and data rate constrant, each transmtter recever par adjusts ts power allocaton accordng to the constraned optmal soluton gven n Secton III. B. Each transmtter recever par also calculates ts correspondng optmal target SINR γ k on the constraned optmal soluton. based 3 Multuser Dstrbuted Power Control Through a control channel, each transmtter acqures the measured SINR γ k (t from the desgnated recever. If γ k If γ k (t γk, the transmsson power wll be updated accordng to (26. (t γk ɛ,, where ɛ s an arbtrary small postve number, the power control algorthm converges to a unque equlbrum pont. Otherwse, t s nfeasble to accommodate all the new users n the current tme slot. The detaled flow chart of the entre procedures of the proposed dstrbuted spectrum access and resource allocaton s gven n Appendx D. Durng the power control stage, f the target SINR γ k cannot be mantaned when transmtter hts ts power bound p max, the network s unable to accommodate all the new users. In ths case, a mult-access control (MAC scheme s requred to guarantee the farness Aprl 13, 2008

17 16 TABLE I UNITS OF SYSTEM PARAMETERS Symbols Descrpton Value p r recevng power W p max maxmal power lmt W B Bandwdth of each subcarrer 100KHz T S Duraton of tme slot 10ms σ 2 Power of thermal nose 10 8 among the users. Ths wll be one of our future efforts. V. SIMULATION RESULT In ths secton, we evaluate the performance and convergence of the proposed dstrbuted energy effcent channel selecton and power allocaton algorthm. The proposed algorthm s frstly nvestgated for each ndvdual user to valdate the theoretcal results. The mpact of system parameter settngs on energy effcency s also analyzed. Furthermore, the convergence of the dstrbuted power control scheme for multple new users sharng the same subcarrers s studed. In addton, we demonstrate that the proposed energy-effcent waterfllng algorthm always outperforms the well-establshed rate-adaptve and margnadaptve waterfllng algorthms n terms of network lfetme. Fnally, we compare the proposed dstrbuted allocaton algorthm wth the global optmal soluton for benchmarkng. A. Smulaton Setup In the smulaton, we consder a wreless ad hoc network wth cogntve rado capablty. Specfcally, the parameters of mca2/mcaz Berkeley sensor motes [23] are used here. The sensor motes operate on 2 AA batteres and the output of each battery s about 1.5 volts, mah. 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 power bound for the transmsson power s 50 mw. The entre spectrum s equally dvded nto subcarrers wth bandwdth 100 khz. The duraton of each tme slot T S s assumed to be 10ms n whch L bts need to be transmtted. Thus, the target data rate s assumed to be r tar = L/T S. The thermal nose power s assumed to be the same over all subcarrers and equals to 10 8 W. The system parameters are summarzed n Table I and they are set such that the target data rate s feasble. Aprl 13, 2008

18 17 Fg. 6. Impact of L to Optmal Data Rate B. Performance of Indvdual Resource Allocaton Algorthm For each ndvdual user, we frst nvestgate the mpact of the target data rate on energy effcency. We consder a transmtter recever par wth avalable subcarrer set Γ(L = 18, the requred data rate r tar = L/T S ranges from bps to bps. The squared lne represents the optmal data rate allocaton wth the ncrease of r tar, whle the damond lne represents the requred data rate r tar. It can be observed from Fg. 6 that the optmal rate and power allocaton remans approxmately 2 unchanged gven the channel condtons of the avalable subcarrers as long as r tar < r opt = After the two lnes converge at L opt = bts, the optmal data rate concdes wth r tar,.e., the requred rate can only be obtaned at the cost of lower energy effcency. It s notceable that L opt s an mportant system desgn parameter, and ts optmal value can be pre-calculated gven the channel condtons. Fg. 7 llustrates the effect of L (thus the target data rate r tar for fxed T S on energy effcency. We defne E = ζ L as the energy consumpton per tme slot whch s jontly determned by ζ and L. It s observed that n case 1 wth the ncrease of L, E ncreases lnearly wth respect to L and the energy consumpton per bt remans approxmately unchanged. When the system enters case 3 due to the 2 due to numercal round-off errors Aprl 13, 2008

19 18 Fg. 7. Impact of L to Energy Effcency ncrease of r tar, ζ degrades whch suggests that the requred data rate r tar s satsfed wth the expense of energy effcency. The mpact of the number of avalable subcarrers on energy effcency s plotted n Fg. 8. It s shown that the ncrease of the number of avalable subcarrers (Γ(L mproves energy effcency by provdng more avalable bandwdth. In fact, the total optmal allocated power to satsfy a fxed target data rate s reduced wth the ncrease of Γ(L. It can be seen n Fg. 8 that the dashed crcle lne (whch represents the unconstraned optmal energy consumpton converges wth the constraned energy consumpton (sold crcle lne when the number of avalable subcarrers reaches 28. It mples that when the avalable subacarrers are less than 28, the unconstraned optmal soluton corresponds to case 3 n Secton III-B. The system wll enter case 1 when Γ(L 28. The performance of the proposed energy-effcent waterfllng wth respect to network lfetme (whch s a crtcal metrc for energy constraned CR ad hoc networks s nvestgated. Assumng unform traffc patterns and persstent traffc flow across the network, we defne the network lfetme as T l = E max /(L ζ, where E max s the maxmal energy source of each transmtter. Compared wth rate-adaptve and margn-adaptve waterfllng (for transmttng the same amount of nformaton bts n the network, t s observed n Fg.9 that the proposed scheme outperforms the other two allocaton schemes n terms of network lfetme. As the optmal allocated rate approaches the target data rate, energy-effcent waterfllng wll converges Aprl 13, 2008

20 19 Fg. 8. Impact of Number of Avalable Subcarrers to Energy Effcency Fg. 9. Performance Comparson Among Dfferent Allocaton Schemes Aprl 13, 2008

21 20 Fg. 10. Convergence of Dstrbuted Power Control wth margn-adaptve waterfllng as expected. However, snce the target data rate n a typcal energy constraned ad hoc network s usually low, t s expected that the proposed scheme wll mprove network lfetme n most applcatons. C. Performance Evaluaton for Multuser Allocaton Scheme After each new user obtans ts optmal subcarrer selecton and power allocaton ndependently, dstrbuted power control (26 may be trggered to manage the co-channel nterference f multple new users happen to choose the same subcarrers. The convergence of allocated power s shown n Fg. 10 (ncludng the total requred power and the power allocated on two randomly chosen subcarrers of two randomly chosen Tx-Rx pars. It s observed that the convergence occurs n 3-4 steps. In ths part of the smulaton (Fg. 11, the performance of the proposed dstrbuted scheme s compared wth the centralzed optmal soluton [25], where t s assumed that a central controller collects all the M N 2 channel gan nformaton from all the N new users, and calculates the global optmal soluton by consderng all the co-channel nterference. Because the centralzed optmal soluton requres solvng M N nonlnear equatons smultaneously, only the 2 2 case s attempted here. It s observed that the proposed dstrbuted scheme (the upper two lnes performs closely to the centralzed optmal soluton (the mddle lne. In Aprl 13, 2008

22 21 Fg. 11. Performance Comparson between dstrbuted scheme and global optmalty addton, the compettve optmal soluton s also shown n Fg. 11, where each user calculates ts own soluton wthout consderng co-channel nterference (thus optmstc. VI. RELATED WORK The mult-user resource allocaton problem based on mult-carrer modulaton such as Orthogonal Frequency Dvson Multplexng (OFDM, where subcarrer band, data rate and power are adaptvely allocated to each user, has been wdely addressed for cellular systems [35], [36]. In mult-carrer drectsequence CDMA (DS-CDMA cellular system, a non-cooperatve power control game for resource allocaton wth respect to maxmzng the energy effcency s proposed n [24] whch leads to the best subcarrer selecton scheme by assumng the realzed SINR on each subcarrer s the same. It s assumed n these works that the spectral utlzaton nformaton s known as a pror wth the ad of base statons, whch s not realstc n scenaros where an nfrastructure s not avalable. Furthermore, t worth notng that the optmal soluton of energy effcent resource allocaton s not best subcarrer selecton for multple transmttng recevng pars n ad hoc networks [25]. In [11], the resource allocaton problem s explored for OFDMA-based wreless ad hoc network by drectly adoptng dstrbuted power control scheme for the power and bts allocaton on all subcarrers Aprl 13, 2008

23 22 to mprove power effcency. A greedy algorthm s proposed for subcarrer selecton n CR networks employng multcarrer CDMA [38], and dstrbuted power control s performed thereafter to resolve co-channel nterference. An Asynchronous Dstrbuted Prcng (ADP scheme s proposed n [37], where the users need to exchange nformaton ndcatng the nterference caused by each user to others. In the context of CR enabled wreless sensor network (WSN [12], a two-step algorthm s proposed to tackle the allocaton problem: channel assgnment wth objectve of mnmzng transmsson power and channel contenton to reserve the subcarrer set for transmsson by ntended transmtters, whle the nterference spectrum mask s assumed to be known a pror. The authors of [13] address the opportunstc spectrum access (OSA problem n WSN, n whch a dstrbuted channel allocaton problem s modeled by a partally observable Markov decson process framework (POMDP whle assumng the transton probablty of each channel s known. In [29], the CR spectrum sharng problem s formulated n mult-hop networks wth objectve to mnmze the space-bandwdth product (SBP. However, the transmsson power allocated on each subcarrer s assumed to be the same whch may lead to sgnfcant performance loss. The effect of power control s analyzed n a subsequent paper [30]. Dynamc Frequency Hoppng Communty (DFHC s proposed n [31] for the spectrum sharng n CR based IEEE wreless regonal area networks (WRANs to ensure QoS satsfacton and relable protecton to lcensed users. In sngle-user scenaro, the optmal allocaton strategy wth objectve to mnmze power or maxmze throughput s named margn-adaptve and rate-adaptve waterfllng over frequency channels [28], respectvely. Whereas, n mult-user case, teratve-waterfllng (IWF s presented n [16] as a dstrbuted scheme for resource allocaton wth the objectve to enhance the throughput. In general, the proposed scheme (ndvdual optmal subcarrer selecton and power allocaton plus mult-user dstrbuted power control provdes a fully dstrbuted but sub-optmal soluton to the orgnal constraned optmzaton problem. Motvated by teratve waterfllng (IWF algorthm n [16], another dstrbuted soluton may be obtaned by solvng the mult-user dstrbuted channel and power allocaton problem teratvely. However, each user can only detect nterference from other users after everyone start transmttng data. It may take many steps for the teratve algorthm to converge f t converges at all, and the delay may be too large. In addton, the cost of the addtonal computatonal complexty may be hgh. Hence, we beleve that the proposed dstrbuted channel allocaton plus power control scheme provdes an effcent and practcal soluton for dynamc spectrum access n CR wreless ad hoc networks employng OFDMA. In addton, t s demonstrated that the proposed scheme performs tghtly to the optmal soluton n the smulatons. Aprl 13, 2008

24 23 VII. CONCLUSION In ths paper, a framework of dstrbuted energy effcent resource allocaton s proposed for energy constraned OFDMA-based cogntve rado wreless ad hoc networks. A mult-dmensonal constraned optmzaton problem s formulated by mnmzng the energy consumpton per bt over the entre avalable subcarrer set for each ndvdual user whle satsfyng ts QoS constrants and power lmt. A two-step soluton s proposed by frst decouplng t nto an unconstraned problem, and a branch and bound method s appled thereafter to obtan the constraned optmal soluton by parttonng the soluton space accordng to power and rate constrants. Co-channel nterference may be ntroduced by concurrent new users and the dstrbuted power control scheme may be trggered to manage the nterference and reach the equlbrum pont n the multuser envronment. The proposed spectrum sharng plus resource allocaton scheme provde a practcal dstrbuted soluton for a CR wreless ad hoc network wth low computatonal complexty. It s mportant to pont out that the proposed algorthm for CR networks can be easly modfed and appled to mult-channel mult-rado (MC-MR networks whch can be consdered as a specal case of the CR based wreless networks [29]. In ths work, t s assumed that the subcarrer detecton s perfect. The effects of detecton errors wll be nvestgated n our future work. ACKNOWLEDGMENT Ths research work s supported n part by the U.S. Army Research Offce under Cooperatve Agreement No. W911NF The vews and conclusons contaned n ths document are those of the authors and should not be nterpreted as representng the offcal polces, ether expressed or mpled, of the Army Research Offce or the U. S. Government. VIII. APPENDIX A The unstraned optmzaton problem (8 s f(ˆp, α := ˆp k + p r k L ( (27 log α k ˆp k k L The frst order dervatve of (27 wth respect to ˆp k can be derved as Aprl 13, 2008

25 24 f(ˆp, α ˆp k = Φ(ˆp, α = ( 1 Φ(ˆp, α log e 2 ˆp k ˆp k + l L,l k ln ( 1 + α k ˆpk + pr + ˆp l ( (28 ln 1 + α ˆp k If k l, c (ˆp l s taken as constant wth respect to ˆpk l L l k s not consdered n ths work. Therefore, (28 can be expressed as snce the mutual nterference between subcarrers Φ(ˆp, α ˆp k = k L ck (ˆpk ln 2 ( ( α ˆp k + p r k 1 + α k k L ˆpk [ ( ] 2 ( α k ˆp k k L ln We assume the data rate k L c k (ˆpk 0 n ths work, thus for f(ˆp,α ˆp k = 0, (29 can be reduce to k as α k 1 + α k ˆpk = k L ln ( 1 + α k ˆp k + p r k L ˆp k From (30, we can derve the unconstraned optmal power allocated for transmtter over subcarrer (30 ˆp k = ˆp k + p r k L ( α k ˆp k α k k L ln (31 From the defnton of unconstraned energy consumpton per bt ˆζ, the frst term of (31 s n the smlar type of ˆζ. If we assume the optmal soluton of (A1 does exst (the subcarrer condton resdes n the feasble regon, there must be a correspondng optmal value of energy per tme slot ˆζ respect to ˆp. Then (31 can be expressed n terms of ˆζ as wth = log e 2 ˆζ 1 α k (32 Aprl 13, 2008

26 25 IX. APPENDIX B The proof of Lemma 1 s gven n ths secton. We frst defne f( p as f( p = Where k K c k = Γ(K log 2 (1 + k K log 2 p + k K p ( Γ(K log e 2 ˆζ 1 + α k + p r. We denote + c k k K (33 h( p = p k K c k = p /Γ(K ( log p (34 /Γ(K log e 2 ˆζ Due to x ln(1 + x x 0, we can obtan ln 2 p /Γ(K ˆζ ln ( 1 + p /Γ(K ˆζ (35 Snce p, Γ(K, and ˆζ 0 n ths work, we get h( p = p /Γ(K ( log p ˆζ (36 /Γ(K log e 2 ˆζ Based on the defnton of f( p, summng h( p and unconstraned optmal energy per bt (ˆζ, we have f( p ˆζ, p R + (37 Thus, we can conclude the ncreasng power deterorates the energy effcency. We defne f( p as X. APPENDIX C Where k K c k = Γ(K log 2 (1 p + p r k K f( p = ( log α k + c k k K k K p Γ(K log e 2 ˆζ. We denote (38 Aprl 13, 2008

27 26 h( p = p k K c k = p /Γ(K ( log 2 1 p (39 /Γ(K log e 2 ˆζ Accordng to (10, = log e 2 ˆζ 1/αk, (39 can be expressed as h( p = p /Γ(K log 2 1 p /Γ(K + 1 α k (40 Due to x ln(1 x, 0 < x < 1, we can obtan ln 2 p /Γ(K ˆζ ln ( 1 p /Γ(K ˆζ, 0 < p /Γ(K < 1 (41 It s reasonable to assume p /Γ(L < 1 due to the maxmal power lmt of energy constraned user (e.g. wreless sensor nodes. Snce p, Γ(K, and ˆζ 0 n ths work, we get h( p = p /Γ(K ( log 2 1 p ˆζ (42 /Γ(K log e 2 ˆζ Based on the defnton of f( p, summng h( p and unconstraned optmal energy per bt (ˆζ, we have f( p ˆζ, p R + (43 Thus, we can conclude the decrease of power wll degrade the energy effcency. Aprl 13, 2008

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