Robust Power and Subcarrier Allocation for OFDM-Based Cognitive Radio Networks Considering Spectrum Sensing Uncertainties

Transcription

1 8 H. FATHI, S. M. S. SADOUGH, ROBUST POWER AD SUBCARRIER ALLOCATIO FOR OFDM-BASED COGITIVE RADIO... Robust Power and Subcarrer Allocaton for OFDM-Based Cogntve Rado etworks Consderng Spectrum Sensng Uncertantes Houshang FATHI, Seyed Mohammad-Sajad SADOUGH Cogntve Telecommuncaton Research Group, Department of Electrcal Engneerng, Faculty of Electrcal and Computer Engneerng, Shahd Behesht Unversty G. C., , Tehran, Iran. Emal: s sadough@sbu.ac.r Abstract. In ths paper, we address power and subcarrer allocaton for cooperatve cogntve rado (CR networks n the presence of spectrum sensng errors. Frst, we derve the mutual nterference of prmary and secondary networks affectng each other by takng nto account spectrum sensng errors. Then, takng nto account the nterference constrant mposed by the cogntve network to the prmary user and the power budget constrant of cogntve network, we maxmze the achevable data rates of secondary users. Besdes, n a mult secondary user scenaro, we propose a suboptmal but low complexty power and subcarrer allocaton algorthm to solve the formulated optmzaton problem. Our numercal results ndcate that the proposed power loadng scheme ncreases the cogntve achevable data rates compared to classcal power loadng algorthms that do not consder spectrum sensng errors. Keywords Cogntve rado (CR, mperfect spectrum sensng, robust algorthm, subcarrer allocaton, power allocaton.. Introducton The rado spectrum s nherently a scarce resource especally n wreless communcaton networks. Moreover, recent studes have shown that the spectrum s not used optmally and spectrum scarcty s more due to neffectve polces n assgnng the spectrum that restrcts ts use solely to lcensed users. A promsng approach to solve the spectrum scarcty s cogntve rado (CR technology that proposes to dynamcally allocate the spectrum to users. In CR, secondary users should constantly montor a predefned frequency band allocated to lcensed prmary n order to detect vacant frequency opportuntes, commonly referred to as spectrum holes, where ths operaton s called spectrum sensng [, [2. Obvously, n practce, durng the spectrum sensng process, t s essental for secondary users to relably detect the prmary user s sgnal n order to avod nterference from the secondary transmsson to the prmary network. However, due to envronmental condtons and transmsson mparments, the spectrum sensng process s an mperfect process,.e., ts results have some uncertantes. The Federal Communcaton Commsson (FCC has suggested geo-locaton and database access as an alternatve to conventonal spectrum sensng for TV band devces (TVBD to access the avalable channels. However, conventonal spectrum sensng s stll needed for an optmal usage of the rado spectrum n future applcatons as suggested by the FCC [3. Optmal power allocaton n practcal CR deployment, ncreases the transmsson capacty of the network and optmzes the power consumpton. More precsely, tradtonal methods proposed for power allocaton, do not consder the coexstence of secondary and prmary networks, and hence, these methods mpose an ntense nterference to prmary users [4, [5. The nterference from secondary user mposed on the prmary user depends from one sde on the spectral nterval between the prmary and secondary systems, and from another sde on the power allocated to the secondary users. Moreover, n practce, the nterference mposed from secondary users on prmary users should not exceed a predefned threshold. In [6, a CR system based on orthogonal frequency dvson multplexng (OFDM s consdered where n order to reduce nterference of the cogntve network on the prmary network, the authors have suggested to elmnate dynamcally adjacent subbands. Obvously, ths technque reduces the bandwdth effcency. In [7, a power allocaton problem s formulated and solved at the CR. The power allocaton strategy proposed n [8, amed at maxmzng the ergodc capacty or mnmzng the outage probablty of the secondary users. In [9, the authors propose a CR power allocaton algorthm that maxmzes the downlnk capacty of secondary users so that the nterference to the prmary user remans n a tolerable range. However, n ths work, spectrum sensng s assumed deal and the error caused by spectrum sensng s not taken nto account. Smlar CR power allocaton schemes are provded n [, [, [2, [3, based on dfferent theoretcal assumptons, wthout however,

2 RADIOEGIEERIG, VOL. 22, O. 3, SEPTEMBER 23 8 consderng the practcal scenaro where the spectrum sensng block s mperfect and has some sensng errors. In [4, the authors propose to consder spectrum sensng errors n ther power allocaton scheme. However, the system model n [4 s very smplstc snce t assumes only a sngle secondary user (.e., not a secondary network composed of multple CR termnals wthout consderng any form of subcarrer allocaton. In ths work, we assume a CR network composed of multple cooperatve secondary users. Then, we extend and generalze ntal results n [4 by proposng both power and OFDM subcarrer allocaton n a mult secondary user scenaro where the soluton of the problem s much dfferent from a sngle secondary user scenaro. More precsely, we frst derve probablstc parameters correspondng to spectrum sensng errors based on detecton probablty and false alarm probablty, whch characterze the spectrum sensng block. Second, we formulate the effect of spectrum sensng errors and ts mpact on the mutual nterference between the secondary network and the prmary network. The derved nterference parameters are used for optmal power allocaton problem at the CR network. More precsely, we express the power allocaton optmzaton problem so as to maxmze the data rate of the secondary network, provded that the nterference to the prmary user subbands does not exceed a predefned amount. We also propose a suboptmal but low complexty algorthm for solvng the optmzaton problem. The rest of ths paper s organzed as follows. The system model and our man assumptons about the model are presented n Secton 2. Secton 3 descrbes the spectrum sensng methodology. Secton 4 characterzes and derves dfferent knds of nterferences that can occur n our consdered transmsson scenaro. Secton 5 presents the man part of the contrbuton,.e., the proposed CR power loadng algorthm that takes nto account spectrum sensng errors. Secton 6 provdes numercal results and dscussons about the performance obtaned by the proposed power loadng algorthm and fnally, Secton 7 draws our conclusons. 2. System Model and Man Assumptons We consder a downlnk transmsson scenaro composed of one par of prmary transmtter/recever and multple secondary (CR termnals, as depcted n Fg.. The prmary and cogntve transmtters use an OFDM sgnalng scheme wth subcarrers. As shown n Fg. 2, the total avalable bandwdth lcensed to prmary transmsson s equal to B Hz where B = f, and the CRs try to have an opportunstc access to each of the prmary subcarrers va spectrum sensng. Let us denote by h SS k, (k {,...,K} and {,...,} the fadng channel gan between the CR transmtter and the k-th CR recever on the -th subcarrer, by h SP ( {,...,} PS h,l PS h K, l SS h, SS h K, SP h Fg.. Topology of the consdered transmsson scenaro. the channel gan between the CR transmtter and the prmary recever on the -th subcarrer, and by h PS k,l (k {,...,K} and l {,...,} the channel gan between the prmary transmtter and the k-th CR recever on the -th subcarrer. These nstantaneous fadng gans are assumed to follow a Raylegh dstrbuton and assumed perfectly known at the CR transmtter. 3. Spectrum Sensng: An Imperfect Process In CR systems, one of the factors that ncreases the nterference level at the prmary transmtter s errors nduced by spectrum sensng, whch s nherently an mperfect process. In what follows, we establsh the man relatons that let us to take nto account errors occurrng durng the spectrum sensng process for preventng harmful nterference from the secondary network and ncreasng the data rates acheved by the CR network. Sensng the presence of a prmary transmtter nsde the -th frequency subband can usually be vewed as a bnary hypothess testng problem wth hypothess H and H defned as: { H : the prmary s not transmttng n the -th subband, : the prmary s transmttng n the -th subband. H otce that the above two hypotheses denote the actual presence or absence of the prmary n the -th subband. However, n practce, the CR network can only have access to an mperfect estmate of the above hypothess as a result of spectrum sensng. We denote these estmated hypothess Ĥ (for the absence of the prmary sgnal n -th subband and Ĥ (for the presence of the prmary sgnal n -th subband. In the sequel, we assume that the cooperatve CR network has made ts fnal decson about the presence or the absence of the prmary sgnal usng the spectrum sensng unt. As shown n Fg. 2, the total avalable bandwdth s dvded to subbands wth a bandwdth equal to f. Due

3 82 H. FATHI, S. M. S. SADOUGH, ROBUST POWER AD SUBCARRIER ALLOCATIO FOR OFDM-BASED COGITIVE RADIO... Fg. 2. Total prmary bandwdth avalable for an opportunstc access to the CR network. to mperfect spectrum sensng, an exact nformaton about the presence of the prmary sgnal n each subband s not avalable and so the CR user can potentally transmt ts data through each of the subbands. We start by defnng the probablty P P = P = P as:,h H = ( P f a P( H + P,H P ( H + P H P ( H + ( P d P ( H ( where P f a = P H s referred to as false-alarm probablty n -th subband, Pd = P H s referred to as detecton probablty n -th subband. Consderng (, the condtonal probablty α can be defned and derved as: ( α P H Ĥ P( H =,Ĥ = P H P P P ( H = ( P f a P( H. (2 P The probablstc framework defned above for characterzng the spectrum sensng process, wll be used n the sequel for takng nto account sensng naccuraces n the proposed CR power loadng scheme. 4. Interference Analyss 4. Evaluaton of the Interference Introduced by the Prmary s Sgnal to the CR s Sgnal The power densty spectrum of the prmary sgnal after M-pont fast Fourer transform (FFT, can be expressed by the expected value of the perodogram [6 as: E{I M (w} = 2πM π π φ PU (e jw ( sn(w ψm/2 sn(w ψ/2 2 dψ (3 where w represents the frequency normalzed to the samplng frequency and φ PU (e jw s the power densty spectrum of the prmary sgnal. The nterference ntroduced by the l-th prmary subcarrer to the -th CR subcarrer that s allocated to k-th secondary user, denoted as J k,l, can be wrtten as: (n+ J k,l = h PS 2 f k,l 2 E{I M (w}dw (4 (n 2 where n = l,.e., n f s the f dstance n frequency between the -th CR subcarrer and the l-th prmary subcarrer. In what follows, we am at dervng the nterference ntroduced by the l-th prmary subcarrer to the -th CR subcarrer under the assumpton that the spectrum sensng process s mperfect. In other words, the prmary and the CR can potentally transmt smultaneously over the same subcarrer,.e., when the spectrum sensng process makes an ncorrect decson (we may thus have nterference scenaros where = l. We assume that the spectrum sensng process has made ts decson n favor of one of the two hypothess Ĥ (.e., the prmary s absent n -th subband or Ĥ (.e., the prmary s operaton n -th subband. However, dependng on the spectrum sensng decson or Ĥ, the nterference level s dfferent. The am of ths part s to derve and characterze these nterference levels under an mperfect spectrum sensng. More precsely, we derve the prmary network nterference on the secondary network by takng nto account spectrum sensng errors. When the decson of spectrum sensng block s Ĥ, the secondary network can transmt data n -th subband, but f the decson s Ĥ, secondary network wll not transmt data n ths subband. When the decson of spectrum sensng block s Ĥ, one of the followng two cases occurs: The prmary s not present n the -th subband (H. Thus, both the prmary and the secondary wll smultaneously send data on ths subband and the wrong decson of spectrum sensng block causes ntense nterference. Moreover, we should also consder the prmary nterference n other subbands on the secondary n -th subband. The prmary s not present the -th subband (H. In ths case, the spectrum sensng block has taken the rght decson and there s only the prmary nterference n other subband on the secondary n the -th subband. As a result, the average total nterference mposed by the prmary network on k-th secondary operatng n -th subband under mperfect spectrum sensng, denoted J k,, wrtes: J k, = = = l= l=,l l=,l,ĥ J k,l = P [,Ĥ J k,l + P(H,Ĥ J k, P J k,l + P(H Ĥ P J k, l=,l J k,l + ( α J k, where J k, s the nterference caused by smultaneous transmsson of prmary and k-th secondary over the -th subcarrer. (5

4 RADIOEGIEERIG, VOL. 22, O. 3, SEPTEMBER Evaluaton of the Interference Introduced by the CR s Sgnal to the Prmary s Sgnal We assume the transmsson of CR s performed by deal yqust pulse and wth power of q k, n -th subband allocated to the k-th secondary, so the power densty spectrum related to the k-th CR transmsson over the -th subband can be wrtten as [6: φ k, ( f = q k, T s ( snπ f T s π f T s 2 where T s s the symbol duraton, equal to / f. The nterference ntroduced by the -th subband of CR to the l-th subband of prmary can be wrtten as: (n+ I k,l = h SP l 2 2 f φ k, ( f d f (n 2 f = q k, T s h SP l where Θ l s defned as: 2 (n+ 2 f (n 2 f ( snπ f T s 2 d f π f T s = q k, Θ l (6 Θ l T s h SP l 2 (n+ 2 f (n 2 f ( snπ f T s π f T s 2 d f. (7 By followng a smlar approach to the computaton of J k,, we can compute Ĩ k, as: Ĩ k, = P [ l=,l = q k, P [ l=,l I l + ( α I Θ l + ( α Θ = q k, Ī (8 where Ĩ k, s the average total nterference mposed under mperfect spectrum sensng by the CR transmsson over the - th subcarrer on the prmary network. Fnally, the total averaged nterference level mposed for secondary transmsson on the prmary transmsson n the -th subcarrer, denoted Ī, s derved as: Ī P [ l=,l Θ l + ( α Θ. (9 5. Proposed Power and Subcarrer Allocaton Scheme Here, we am at allocatng the power to each subcarrer of the OFDM-based CR so as to maxmze the achevable nformaton rate for the CR network whle keepng the nstantaneous nterference ntroduced to the prmary below a predefned threshold. Consderng an deal codng scheme and usng the Shannon capacty formula, the cogntve achevable data rate at the -th subcarrer that s allocated to the k-th secondary, denoted R k, for {,...,} and k {,...,K}, s gven by: [ R k, (q k, = α f log 2 + hss k, q k, σ 2 k, + ( J k, where q k, s the total transmt power n the -th subcarrer that s allocated to the k-th secondary, and σ 2 k, denotes the addtve whte Gaussan nose (AWG varance. Usng (, the proposed cogntve power allocaton under mperfect spectrum sensng s formulated as the followng optmzaton problem: subject to: C = max K k= = K k= = K q k,,x k, k= = x k, q k, Ī I th, x k, q k, Q, K k= x k,, q k, x k, R k, (q k, k, x k, {,} k ( where C ndcates the transmsson capacty of the CR user, I th s the total tolerable nterference at the prmary and Q s the total maxmum power constrant at the CR network; x k, s a bnary varable ndcatng that subcarrer s allocated to secondary user k or not, whereas each subcarrer can be allocated to only one secondary user. The optmzaton problem ( for K = s a standard convex problem that can be solved n a straghtforward manner by usng the Lagrange multpler method [5. Proposton: In a sngle CR user case (K =, the optmal soluton s derved as: { q = max, σ2 + } J λ + µī h SS 2, =,..., (2 where λ and µ are the Lagrange multplers. Proof: The proof s provded n the appendx. The optmzaton problem ( for the case of multple secondary users s a mxed nteger non-lnear programmng (MILP problem, for whch the optmal soluton s Phard [6. However, n the sequel, we propose a suboptmal algorthm wth low complexty to solve t. Frst, we defne η k, as follows: η k, = hss k, 2 σ 2 k, + J k,,,k. (3

5 84 H. FATHI, S. M. S. SADOUGH, ROBUST POWER AD SUBCARRIER ALLOCATIO FOR OFDM-BASED COGITIVE RADIO... Then, for each subcarrer and CR user k, we calculate η k, and we follow a greedy scheme [7, [8,.e., for the -th subcarrer, the secondary havng the maxmum value η k, s allowed to transmt over ths subcarrer. The ndex of ths user s denoted k,.e., we have: k = argmax η k,, =,...,. (4 k Second, we allocate the power to subcarrers as follows: q k, = Q η k, = η k,, =,..., (5 where Q s defned n Tab., that s the power level that satsfes the constrants of power budget and nterference n (. Obvously, n our algorthm, greater power s allocated to subcarrers havng greater parameter η k,. Complexty ssues of the proposed algorthm: We can fnd the optmal soluton of the problem ( by frst fndng the optmal subband allocaton to the secondary users by an exhaustve search and then the power allocaton can be optmzed by another exhaustve search over subbands. Hence, the complexty of the optmum soluton s O(K. The complexty of the optmal algorthm proposed n [9 s O(K 2. Our proposed suboptmal algorthm has a reduced complexty of O(K. Thus, although suboptmal, the proposed algorthm s less complex n terms of practcal mplementaton. The proposed suboptmal and low complexty algorthm for CR power and subcarrer allocaton s provded below. Proposed power and subcarrer allocaton algorthm Set x k, =, Set q k, =, k k hss k, 2 Calculate η k, = σ 2 k, +, k J k, k = argmax k η k,, =,..., x k, =, =,..., q = Q/ Calculate Ī, =,..., from (9 Calculate Î = q = Ī f Î > I th Set q = Ith = Ī end Set Q = q Set q k = Q η k,, =,...,, = η k, f q k =, =,...,, Set x k, = end Calculate α, =,..., from (2 Calculate C = K k= = α x k, f log 2 [ + qk, η k, Tab.. Pseudo code of the proposed power and subcarrer allocaton algorthm. 6. umercal Results and Dscusson Throughout ths secton, the total avalable bandwdth (B s assumed equal to 2 MHz. We also assume that 2 OFDM subbands are avalable for cogntve transmsson wth a bandwdth equal to MHz for each subcarrer. It s assumed that a long-enough cyclc prefx (CP s employed at the OFDM transmtter. The a pror probabltes about the presence of the prmary n dfferent subbands are gathered n vector P(H = [.75,.6,.7,.2,.5,.25,.,.55,.7,.6,.2,.3. We also assume that the probablty of false alarm s equal to.8 n all subbands and the detecton probablty of subbands s equal to P d = [.97,.94,.96,.98,.95,.99,.98,.97,.96,.95,.98,.99. Moreover, the nose varance s set to dbm and the fadng coeffcents for all channels are assumed to follow a Raylegh dstrbuton wth an average channel power gan of dbm. The CR network has 5 secondery users (K = 5. Maxmum transmtted data rate of CR users (Mbps Maxmum transmtted data rate of CR users (Mbps Proposed algorthm Optmal soluton Interference ntroduced to the prmary user band (Watt Fg. 3. Maxmum acheved data rate of CR user versus the average nterference mposed to the prmary user band Wthout consderng sensng errors Proposed scheme consderng sensng errors Interference ntroduced to the prmary user band (Watt Fg. 4. Maxmum acheved data rate of CR user versus the average nterference mposed to the prmary user band. Fgure 3 depcts the maxmum transmtted data rate of CR user versus the average nterference mposed to the prmary band where the transmt power budget (Q s fxed and set to 3 Watt. In ths fgure, for the sake of comparson, we have provded the data rates acheved by the optmal algo-

6 RADIOEGIEERIG, VOL. 22, O. 3, SEPTEMBER rthm proposed n [9 to obtan the optmal power allocaton of (. Ths fgure shows that the proposed algorthm provdes data rates whch are not very far from those acheved by the optmal soluton n [9. Fgure 4 depcts the maxmum transmtted data rate of CR user versus the average nterference mposed to the prmary user band n whch the transmt power budget s fxed and set to 3 Watt. Ths fgure shows that the proposed scheme leads to hgher CR data rates for a gven nterference level mposed to the prmary user band. Smlar plots are provded n Fg. 5 showng the maxmum transmtted data rate of CR user versus ths tme the total maxmum power constrant of CR network and a fxed predefned nterference threshold s I th =.2 Watt. Agan, we observe the superorty of our proposed method compared to the conventonal method that does not consder errors nduced by spectrum sensng. Maxmum transmtted data rate of CR users (Mbps Wthout consderng sensng errors Proposed scheme consderng sensng errors Maxmum power budget n CR network (Watt Fg. 5. Maxmum acheved data rate of CR user versus the total maxmum power constrant at the CR network. Fgure 6 depcts the power allocaton (q and nterference factor (Ī n CR subbands at a fxed nterference level (I th =.2 Watt and a fxed total power budget (Q = 3 Watt. We observe that by usng our proposed algorthm, a larger amount of power s allocated to subcarrers wth a lower nterference level, and vce versa Allocated Power (n Watts Interference Factor (X Index of subbands Fg. 6. CR power allocaton among OFDM subcarrers, I th =.2 Watt and Q = 3 Watt. 7. Concluson We proposed power and subcarrer allocaton scheme for OFDM-based CR system whle takng spectrum sensng errors nto account. We provded approprate relatons for modelng the mutual nterference between the CR network and the prmary network. We have satsfed the requred QoS of the prmary network by consderng the nterference mposed from the CR to the prmary. Smlarly, we have taken nto account the nterference mposed from the prmary network on the CR network to satsfy the secondary QoS, n terms of achevable rates. Besdes, we propose a suboptmal power allocaton algorthm wth lower complexty to solve the optmzaton problem. It was shown that the rates acheved by usng the proposed algorthm are not very far from rates acheved by the optmal (but not practcal soluton. Moreover, smulaton results confrmed that the proposed scheme maxmzes the data rate acheved by the CR network, whle keepng the nterference mposed on prmary network wthn a tolerable lmt. Appendx Dervaton of q n (2 We start by convertng the optmzaton problem n ( to the standard form [5. To ths end, we wrte the problem as: subject to: mn q = = α f log 2 [ 2 q + hss σ 2 + J (6 q Ī I th, (7 = q Q, (8 q. (9 The problem n (6 s a nonlnear convex problem, n other words, the objectve functon s nonlnear and concave wth respect to q, so, the optmal soluton can be obtaned by the Karush-Kuhn-Tucker (KKT condtons [5. Consderng Lagrange multplers λ, µ and ν for the nequalty constrants n (7, (8 and (9, respectvely, we can defne the Lagrangan L assocated wth (6, (7, (8 and (9 as: L(q,λ,µ,ν = R (q + λ ( + µ ( = = q Ī I th q Q ν q (2

7 86 H. FATHI, S. M. S. SADOUGH, ROBUST POWER AD SUBCARRIER ALLOCATIO FOR OFDM-BASED COGITIVE RADIO... where R (q = = [ α f log 2 + hss 2 q σ 2 +. J Consderng that (23 should be equal or greater than zero, we derve q as: { q = max, (λ + µī η }, (27 ow we express the gradent of L(q,λ,µ,ν wth respect to q as: and ths ends the proof. q L(q,λ,µ,ν = f ln2 + λ Ī + µ ν (2 q + /η where as: η hss 2 σ 2 +. J To solve the problem, we can wrte the KKT condtons = q Ī I th, = q Q, q λ, µ, ν, λ ( q Ī I th =, V µ ( q Q =, V, ν q =, q + /η + λī + µ ν = (22 where λ = ln2 f λ, µ = ln2 f µ and ν = ln2 f ν. Consderng the condton ν q =, we can remove ν from (22 and rewrte t as: = = q, (23 q Ī = I th, (24 q = Q, (25 q + /η = λī + µ. (26 Substtutng q from (26 nto (24 and (25, we get the Lagrange multplers, µ and λ. Acknowledgements The authors would lke to acknowledge the Research Insttute for ICT (ITRC for supportng ths work by allocatng a research grant. References [ HAYKI, S. Cogntve rado: Bran-empowered wreless communcatons. IEEE Journal on Selected Areas n Communcatons, 25, vol. 23, no. 2, p [2 MITOLA, J. Cogntve rado for flexble moble multmeda communcatons. In IEEE Internatonal Workshop on Moble Multmeda Communcatons. San Dego (CA, USA, 999, p [3 Federal Communcatons Commsson. Second Report and Order and Memorandum Opnon and Order, 2. [Onlne. Avalable at: [4 LETAIEF, K. B., ZHAG, Y. J. Dynamc multuser resource allocaton and adaptaton for wreless systems. IEEE Wreless Communcatons, 26, vol. 3, no. 4, p [5 RHEE, W., CIOFFI, J. M. Increase n capacty of multuser OFDM system usng dynamc subchannel allocaton. In Proceedngs of 5 st IEEE Vehcular Technology Conference VTC Sprng. Tokyo (Japan, 2, vol. 2, p [6 WEISS, T., HILLEBRAD, J., KROH, A., JODRAL, F. K. Mutual nterference n OFDM-based spectrum poolng systems. In Proceedngs of 59 th IEEE Vehcular Technology Conference VTC Sprng. Mlan (Italy, 24, vol. 4, p [7 ZHAG, Y., LEUG, C. Resource allocaton n an OFDM-based cogntve rado system. IEEE Transactons on Communcatons, 29, vol. 57, no. 7, p [8 KAG, X., ZHAG, R., LIAG, Y.-C., GARG, H. Optmal power allocaton strateges for fadng cogntve rado channels wth prmary user outage constrant. IEEE Journal on Selected Areas n Communcatons, 2, Vol. 29, no. 2, p [9 BASAL, G., HOSSAI, M., BHARGAVA, V. Optmal and suboptmal power allocaton schemes for OFDM-based cogntve rado systems. IEEE Transactons on Wreless Communcatons, 28, vol. 7, no., p [ ZHAO, G., YAG, C., LI, G. Y., LI, D., SOOG, A. C. K. Power and channel allocaton for cooperatve relay n cogntve rado networks. IEEE Journal of Selected Topcs n Sgnal Processng, 2, vol. 5, no., p [ TIA, Z., LEUS, G., LOTTICI, V. Jont dynamc resource allocaton and waveform adaptaton for cogntve networks. IEEE Journal on Selected Areas n Communcatons, 2, vol. 29, no. 2, p

8 RADIOEGIEERIG, VOL. 22, O. 3, SEPTEMBER [2 BASAL, G., HOSSAI, M., BHARGAVA, V. Adaptve power loadng for OFDM-based cogntve rado systems wth statstcal nterference constrant. IEEE Transactons on Wreless Communcatons, 2, vol., no. 9, p [3 WAG, S. Effcent resource allocaton algorthm for cogntve OFDM systems. IEEE Communcatons Letters, 2, vol. 4, no. 8, p [4 SADOUGH, S. M.-S., KHAJAVI,. T. Power allocaton for OFDMbased cogntve rado systems under mperfect spectrum sensng. Internatonal Journal on Communcatons Antenna and Propagaton (IRECAP, 22, vol. 2, no. 4, p [5 BOYD, S., VADEBERGHE, L. Convex Optmzaton. Cambrdge Unversty Press, 24. [6 FLOUDAS, C. A. onlnear and Mxed Integer Optmzaton: Fundamentals and Applcatons. ew York (Y, USA: Oxford Unversty Press, 995. [7 CORME, T. H., LEISERSO, C. E., RIVEST, R. L., STEI, C. Introducton to Algorthms. Thrd Edton. Cambrdge (MA, USA: MIT Press, 29. [8 DASGUPTA, S., PAPADIMITRIOU, C. H., VAZI- RAI, U. V. Algorthms. 26. [Onlne Avalable at: vazran/algorthms/all.pdf. [9 GU, H., WAG, S., LI, B. Maxmze sum capacty of multuser cogntve OFDM systems. In 2th Annual Wreless and Optcal Communcatons Conference (WOCC. ewark (J, USA, 2, p. - 5.