RECENT years have witnessed a great interest in cognitive

Transcription

1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 11, DECEMBER Prce-Based Jont Beamformng and Spectrum Management n Mult-Antenna Cogntve Rado Networks Dep N. Nguyen and Marwan Krunz Abstract We consder the problem of maxmzng the throughput of a mult-antenna cogntve rado (CR) network. Wth spatal multplexng over each frequency band, a multantenna CR node controls ts antenna radaton drectons and allocates power for each data stream by approprately adustng ts precodng matrx. Our obectve s to desgn a set of precodng matrces (one per band) at each CR node so that power and spectrum are optmally allocated for the node and ts nterference s steered away from unntended recevers. The problem s nonconvex, wth the number of varables growng quadratcally wth the number of antenna elements. To tackle t, we translate t nto a noncooperatve game. We derve an optmal prcng polcy for each node, whch adapts to the node s neghborng condtons and drves the game to a Nash-Equlbrum (NE). The network throughput under ths NE equals to that of a locally optmal soluton of the non-convex centralzed problem. To fnd the set of precodng matrces at each node (best response), we develop a low-complexty dstrbuted algorthm by explotng the strong dualty of the convex per-user optmzaton problem. The number of varables n the dstrbuted algorthm s ndependent of the number of antenna elements. A centralzed (cooperatve) algorthm s also developed. Smulatons show that the network throughput under the dstrbuted algorthm rapdly converges to that of the centralzed one. Fnally, we develop a MAC protocol that mplements our resource allocaton and beamformng scheme. Extensve smulatons show that the proposed protocol dramatcally mproves the network throughput and reduces power consumpton. Index Terms Noncooperatve game, prcng, cogntve rado, MIMO, power allocaton, frequency management, beamformng. I. INTRODUCTION RECENT years have wtnessed a great nterest n cogntve rado (CR) and mult-nput mult-output (MIMO) technologes. Through spectrum sensng, CRs can opportunstcally communcate on temporarly dle frequency bands whle avodng nterference wth lcensed prmary users (PUs). MIMO communcatons mprove the lnk throughput by sendng ndependent data streams smultaneously over dfferent antennas (a.k.a. spatal multplexng). A crucal challenge n CR networks (CRNs) s how to effectvely allocate transmsson powers and spectrum among CRs (see Fg. 1(a)) so as to maxmze the network throughput whle protectng prmary users (PUs) from CR nterference. Manuscrpt receved 15 December 2011; revsed 1 June Ths research was supported n part by NSF (under grants CNS , CNS , IIP , and IIP ), Raytheon, and the Connecton One center. Any opnons, fndngs, conclusons, or recommendatons expressed n ths paper are those of the author(s) and do not necessarly reflect the vews of the Natonal Scence Foundaton. The authors are wth Department of Electrcal and Computer Engneerng, Unversty of Arzona (e-mal:{dnnguyen, krunz}@emal.arzona.edu). Dgtal Obect Identfer /JSAC /12/$31.00 c 2012 IEEE Even for a sngle channel and sngle-antenna wreless devces, the problem s dffcult due to the non-convexty of the network throughput functon. The ncorporaton of MIMO technques nto CR systems ntroduces two new control dmensons, besdes power control and frequency management: power allocaton over antennas (space dmenson) and nterference management. The latter comes from MIMO s degrees of freedom [1], whch allow a MIMO node to suppress nterference from others and beamform ts antenna patterns to keep nterference away from unntended recevers. MIMO s power allocaton and nterference management can be ontly controlled va precodng matrces, a spatal multplexng technque [1]. Prevous works (e.g., [2] [3] [4]) consdered power allocaton or stream control (see Fg. 1(b)), but dd not take nto account nterference management va controllng the antenna beams. An optmal set of precodng matrces for each node allocates power over both space and frequency dmensons (Fg. 1(c)) and yelds radaton patterns that nduce mnmum nterference (Fg. 1(d)), so as to maxmze network throughput. Ths problem s the focus of our work. II. RELATED WORKS Ignorng the need to protect PUs, the ntegraton of MIMO nto CRNs very much resembles mult-carrer (e.g., OFDM) MIMO (MC-MIMO) systems. In MC-MIMO, ont power and spectrum optmzaton s a non-convex problem, whch was recently shown to be NP-hard [5],.e., ts complexty grows exponentally wth the number of varables. Unfortunately, the number of varables n a MC-MIMO network can be very large. For nstance, n a network of 10 lnks, 4 antennas per node, and 10 sub-carrers, the problem nvolves = 1600 complex varables (or 3200 real varables). Exstng works on MIMO CR systems (e.g., [4] [6] [7] [8] [9]) generally overlook the optmzaton over the frequency dmenson. Extendng these works to mult-band MIMO CRNs s not trval. Frst, scalar-value algorthms used for a sngleband MIMO ad hoc network (e.g., bsecton search n [9]) do not work when searchng for optmal vectors n mult-band MIMO CRNs. Second, as shown n ths paper, even wthout beamformng the prce-based optmal power allocaton over both frequency and spatal dmensons s not equvalent to a general water fllng problem (wth multple water levels) [10]. Hence, exstng algorthms for MIMO (e.g., [10] [11] [12] [13]) and SISO systems (e.g., [14]) are not applcable. Thrd, applyng sngle-band MIMO technques to each ndvdual band of a mult-band MIMO CR system often leads to poor performance, as shown later n ths paper.

2 2296 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 11, DECEMBER 2012 (a) (c) (d) Fg. 1. Power allocaton (a) n frequency, (b) n space, (c) n both dmensons, and (d) four transmt radaton patterns steerng away from nearby unntended recevers. In [4] [8], a sngle-frequency cogntve MIMO network was formulated as a noncooperatve game. In these works, the spectrum management and beamformng capablty of CR MIMO transmtters were not taken nto account. Moreover, prcng [15] was not used n [4] [8], hence a node maxmzes ts throughout n a greedy manner. Smulatons show that our prcng technque greatly outperforms greedy mechansms n terms of network throughput and power effcency. Relyng on varatonal nequalty theory, matrx proecton, and fxedpont theorem, the authors n [4] [8] requre the channel gan matrces among CR nodes to meet certan condtons so that the exstence and convergence of the NE are guaranteed. In other words, the NE exstence and the convergence property of ther dstrbuted algorthm depend on gven channel condtons that are not always met. Motvated by the above, ths paper develops a lowcomplexty dstrbuted algorthm that confgures the transmt antenna radaton drectons and allocates powers for all data streams so as to maxmze the network throughput. Our man contrbutons are as follows. Frst, we formulate the ont power, spectrum allocaton, and beamformng problem as a noncooperatve game [16]. We prove that the game always admts at least one NE and we provde condtons for the unqueness of the NE. Second, to mprove the NE effcency, we derve userdependent prcng polces that drve the game to a NE whose network throughput s the same as that of a locally (b) optmal pont of the nonconvex network-wde problem. By capturng the nterference from a transmtter to ts unntended recevers, the prcng polcy gudes the MIMO transmtters to steer ther beams away from nearby unntended recevers. Va smulatons, we observe that the proposed prcng polces dramatcally mprove the network throughput over the greedy approach (whch does not use prcng). Our approach s also more power-effcent than the greedy approach where users use all of ther power budgets to greedly maxmze ther ndvdual throughput. Thrd, explotng the strong dualty n convex optmzaton, we desgn a low-complexty dstrbuted algorthm that allows a node to compute ts set of precoders (best response) n constant tme w.r.t. the antenna array sze. We also develop a centralzed algorthm for the network optmzaton problem, where nodes are assumed to work n a cooperatve way (cooperatve game). Smulatons show that the performance of the (noncooperatve) dstrbuted algorthm s almost the same as that of the (cooperatve) centralzed one. Forth, we desgn full/generalzed egen MIMO precodng for mult-channel systems. Ths dffers from a large body of works on MIMO precoder desgn (e.g., [6] [17] [18]), where only one data stream s sent from a MIMO transmtter on a sngle channel. In these works, precoders have a rank of one and reduce to vectors. In generalzed egencodng, there s no constrant on the rank of the precodng matrces [19],.e., several data streams can be sent smultaneously. Inspred by the ntroducton of MIMO spatal multplexng nto exstng networks (e.g., n allows up to four concurrent multplexed streams), generalzed egencodng has recently attracted great nterest. In Secton III, we present the network model and the problem formulaton. The noncooperatve game analyss, optmal prcng polcy, convergence proof, the dstrbuted algorthm, and a correspondng MAC protocol are gven n Secton IV. The centralzed algorthm s developed n Secton V. Numercal results are dscussed n Secton VI. Concludng remarks and future work are provded n Secton VII. Throughout the paper, we use (.) to denote the conugate matrx, (.) H to denote the Hermtan matrx transpose, tr(.) for the trace of a matrx, det(.) for the determnant, and (.) T for matrx transpose. Matrces and vectors are bold-faced. III. PROBLEM FORMULATION We consder a CRN that coexsts wth several prmary networks. The CRN conssts of N CR lnks. Each CR node s equpped wth M antennas. The avalable spectrum conssts of K orthogonal frequency channels wth central frequences f 1, f 2,..., f K. For smplcty, we also use f k to refer to the kth channel. Let Φ N = {1, 2,...,N} and Ψ K = {1, 2,...,K} denote the sets of CR lnks and channels, respectvely. Each CR user can smultaneously communcate over multple channels. We mpose a half-duplex constrant on all transmssons. On a gven channel, a CR transmtter can send up to M ndependent data streams on ts M antennas. Formally, for channel f k let x be a column vector of M nformaton symbols, sent from node to ts destnaton node d(). Each element of x s from one data stream. Let C M M denote the complex precodng matrx of node on channel f k. Then, the transmt vector s x. We assume spectrum

3 NGUYEN and KRUNZ: PRICE-BASED JOINT BEAMFORMING AND SPECTRUM MANAGEMENT IN MULTI-ANTENNA COGNITIVE RADIO NETWORKS 2297 sharng among dfferent CRs. Specfcally, for channel f k,the receved sgnal vector y d() at recever d() of lnk (, d()) s gven by: y d() = H d(), x + H d(), x + N k {Φ N \} (1) where the frst term n the RHS of (1) s the desred sgnal sent from transmtter. H d(), s an M M channel gan matrx on channel f k from the transmtter to d(). Each element of H d(), s a multplcaton of a dstance- and channeldependent attenuaton term and a random term that reflects mult-path fadng (complex Gaussan varables wth zero mean and unt varance). We assume a flat-fadng channel. The second term n (1) represents nterference from other CR lnks that lnk (, d()) shares the channel f k wth. N k C M s an M 1 complex Gaussan nose vector wth dentty covarance matrx I, representng the floor nose plus normalzed (and whtened) nterference from PUs on channel k. The Shannon rate of lnk (, d()) on channel f k s [1]: R =logdet(i + d(), C d() H d(), ) (2) where C d() s the nose-plus-nterference covarance matrx at d() over channel f k, gven by: C d() = I + {Φ N \} H d(), H H d(),. The total channel rate over all frequency bands of lnk s: R =. (3) R We use P () to denote the power allocated on channel k (frequency dmenson) at antenna s (space dmenson) of CR user. For user, the total power allocated on all frequency bands and all antennas should not exceed a maxmum power budget P max. Consequently, M s=1 P () = ) P max. (4) PU protecton s provded n the form of databaseauthorzed access [20] and frequency-dependent power masks on CR transmt powers. Note that the FCC [20] recently mposed power masks even for dle channels, f such channels are adacent to PU-actve channels. Let P mask = (P mask (f 1 ),P mask (f 2 ),...,P mask (f K )) denote the power mask on all channels, we requre: M s=1 P () = tr( ) P mask (f k ). (5) Mathematcally, the network throughput maxmzaton problem can be stated as follows: maxmze R {,, Φ N } Φ N s.t. C1: C2: ) P max, Φ N ) P mask (f k ), k Ψ K, Φ N. (6) IV. GAME THEORETIC DESIGN The optmzaton problem n (6) s not convex. Thus, even a centralzed computaton of the globally optmal soluton s prohbtvely expensve. To develop a dstrbuted algorthm, we reformulate (6) as a noncooperatve game and derve a prcng functon for each CR lnk that guarantees a locally optmal soluton. A. Game Formulaton A noncooperatve game s characterzed by a set of players, ther acton/strategy spaces, and correspondng utlty/payoff functons. For the above CRN, the set of CR lnks Φ N represents the set of players. The acton space s the unon of the acton spaces of varous players, subect to constrants C1 and C2 n (6). The acton/strategy space for each player s the set of all possble precodng matrces for the K frequency channels n Ψ K. Formally, an acton from the acton space of lnk s denoted by = ( {1}, {2},..., ), whch can be vewed as an M KM block matrx, comprsed of KM M matrces. Let =( 1, 2,..., 1, +1,..., N ) be the set of actons from all lnks, except lnk. The utlty or payoff of player for ts acton s mapped to lnk s Shannon rate, whch also depends on the selecton of precodng matrces from other CR lnks : U (, ) = R. (7) Due to the noncooperatve nature of the game, the transmtter of each lnk allocates ts transmsson power over both space and frequency dmensons, and confgures ts radaton pattern to maxmze ts own return. Formally, each CR user solves the followng problem for ts set of precodng matrces : s.t. maxmze {, } C1 : C2 : U (, ) ) P max ) P mask (f k ), k Ψ K. By solvng the above problem, CR users mplctly nteract wth each other through ther choces of the precodng matrces. Under some condtons, the game reaches a NE where no user has an ncentve to unlaterally devate from. However, as each CR user behaves selfshly, the resultng NE s often far from the Pareto optmum, and the network throughput can be low. The effcency of the NE can be mproved by usng approprate prcng polces [15]. The utlty functon wth prce s ned as: (8) U (, ) = U (, ) F ( ) (9) where F ( ) s the prcng functon for lnk. Consequently, we come up wth the followng noncooperatve game wth prcng n whch each player Φ N solves the followng problem: maxmze U { (, ) } (10) s.t. C1 and C2 as n problem (8). B. Prcng Polcy In economcs, the prcng functon can take varous forms to account for varous marketng and prcng polces, e.g.,

4 2298 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 11, DECEMBER 2012 volume dscount, coupon dscount, etc. In the context of network resource allocaton, both lnear (e.g., [14] [21]) and nonlnear [22] prcng functons have been proposed to acheve one of two purposes. Frst, prcng has been used to mpose desrable constrants by adustng the cost/prce of volaton. These constrants can be, for examples, transmsson rate demands [23] or nterference constrants on CR transmssons [24]. Second, prcng has been used to mprove the effcency of a NE of a noncooperatve game [14]. Prcng dscourages players from behavng selfshly and ncentvzes them to work n a cooperatve way [15]. In ths case, actons are not free or equally expensve. Players have to pay dfferent taxes or prces for dfferent actons, based on the level that these actons adversely affect the socal welfare. In ths work, prcng serves the second purpose. We ne the prcng functon F u ( ) as follows: F ( )=tr [ H A ] (11) where A = A (2) A (K) A (1) (12) s a KM KM block dagonal matrx, consstng of K blocks along ts dagonal. The kth block A s an M M postvesemnte matrx. A s referred to as the prcng-factor matrx of CR lnk and A s referred to as the prcngfactor submatrx at channel k of lnk. The followng theorem guarantees the exstence of a NE of the game (10). Theorem 1: There exsts at least one NE for the noncooperatve game n (10). Proof: See [25]. To guarantee a lower bound on the effcency of the acheved NE, we next derve a user-dependent prcng functon. The proposed prcng polcy ensures that at the resultng NE, the CRN throughput s at least as good as that of a locally optmal soluton to the network optmzaton problem (6). Theorem 2: Let the kth matrx A of the block dagonal prcng-factor matrx A n (12) be set to: A = d(), C d() H d(), [( ) 1 + {Φ N \} (13) d(), C d() H d(), ] 1 d(), C d() H d(),. Then, the CRN s throughput at a NE of the game (10) equals to that of a locally optmal soluton of the network-wde problem (6). Proof: See Appendx A. The ratonale behnd the choce of prcng functon n (11) s to facltate the dervaton of the prcng factor matrx n Theorem 2. To gve a physcal nterpretaton of the prcng functon, consder a specal case where a CR node uses omndrectonal transmsson and equally allocates power on all frequency bands. In such a case, the precoders are dagonal matrces wth dentcal dagonal elements. Hence, the prcng functon n (11) s a weghted functon of the powers P () allocated on streams (s, k). The weghts are exactly the dagonal elements (s, k) of the prcng factor matrx A that captures the per-unt prce of possble nterference on that spatalspectrum drecton. Hence, the prcng functon captures the nterference that a transmtter nduces on unntended recevers for a gven set of precoders. It s worth notng that the perunt prce of nterference depends on the set of unntended recevers. In other words, the nterference prce vares from one market (user) to another. To compute the prcng-factor matrx A n (13), a CR transmtter needs to obtan feedback regardng the nterferenceplus-nose covarance, and the precodng and channel matrces from other lnks. In practce, f the channel gan matrx from to d() s weak,.e., H d(), 0, there s no need for d() to send ts feedback to. Hence, only gets feedback from recevers d() that are wthn s vcnty. It s also worth notng that the feedback nformaton s locally avalable at a recever d() as a byproduct of ts decodng process (e.g., successve nterference cancelaton (SIC) recevers [1]). The kth block A of the prcng factor matrx n (13) s smlar to that n [9] for a sngle-band MIMO ad hoc network usng frst-order Taylor seres approxmaton. C. Best Response: Optmal Antenna Radaton Drectons and Power Allocaton We now solve the ndvdual utlty optmzaton problem (10), from whch a CR user fnds ts best response gven others actons. Because problem (10) s convex, t can be solved by standard methods, e.g., nteror pont [26], requrng polynomal tme w.r.t. to the number of varables. In [6], the authors solved a smlar problem usng semnte programmng. However, the number of varables n (10) grows quadratcally wth the number of antennas, and can be very large. In ths secton, we develop an effcent algorthm whose complexty s ndependent of the number of antennas. Recallng the convexty of (10) and that the Slater s condtons can easly be shown to hold [26], strong dualty holds for problem (10),.e., an optmal soluton to (10) should also solve the followng dual problem: DP : mnmze {α,γ 0, } D(α,γ ) (14) where D(α,γ ) s the dual functon, ned as: D(α,γ )= max {, } In the above, L (,α L (,α,γ ). (15),γ ) s the Lagrangan functon of the utlty maxmzaton problem at user, wrtten as (16), where α and γ are nonnegatve Lagrangan multplers. Theorem 3: The M KM block matrx that solves (10) (for the user s best response) must have ts kth block, n a form of a generalzed egen matrx of the matrces d(), C d() H d(), and A +(α + γ )I, whereα and γ are the optmal Lagrange multplers of (10). In other words, the followng equatons must hold k Ψ K M M dagonal matrx Λ : d(), C d() H d(), =[A +(α + γ )I] for a Λ. (20) Proof: See Appendx B. As dscussed before, the precodng matrx determnes both the drectons of the antenna radaton as well as how

5 NGUYEN and KRUNZ: PRICE-BASED JOINT BEAMFORMING AND SPECTRUM MANAGEMENT IN MULTI-ANTENNA COGNITIVE RADIO NETWORKS 2299 L (,α,γ )=U (, ) α = tr( A L (,α,γ )= D(α,γ )= {R { s=1 { s=1 M {log(1+p () [ dag s (Π )} ) P mask (f k )] γ [ ) P max ] α )) P () [ dag s (Ω M {log dag s (Π ) dag s (Ω ) 1+ dag s (Ω ) dag s (Π ) } + α s, k such that dag s (Π ) > dag s (Ω ) > 0. L(α,γ,p,λ) =D(α,γ )+ p 2 {(max{0,λ 1 pγ }) 2 (λ 1 ) 2 }+ p 2 ) P mask (f k )] γ [ ) P max ] (16) )}+α P mask (f k )+ γ K P max} (17) P mask (f k )+ γ K P max} (18) {(max{0,λ k+1 pα }) 2 (λ k+1 ) 2 } (19) node allocates ts transmsson power on dfferent antennas over channel k. Theorem 3 provdes a class of matrces that the solutons of (10) must belong to. Ths class gves the drectons that user should pont ts antenna radaton to. The next step s to fnd the optmal power allocaton P () over the set of KM data streams. To ensure that belongs to the class of matrces specfed by Theorem 3, let: P () 1/2 k = T (21) where T s an M M matrx wth unt-norm column vectors that satsfes (20). Ths matrx can be found by normalzng the generalzed egen matrx. P () 1/2 k s a square root matrx of the M M dagonal matrx P () k whose dagonal entry (s, s) s the power allocated for sub-channel (s, k), P (). We can verfy that the expresson of n (21) satsfes (20). As s a generalzed egen matrx of matrces d(), C d() H d(), and A +(α + γ )I, T should dagonalze each of the two matrces [27]: T H [ d(), C d() H d(), ]T = Π and (22) T H [A +(α + γ )I]T = Ω where Π and Ω are M M dagonal matrces. Note that although ts columns have unt-norm, T s not an orthonormal matrx, as A s generally not smlar to I. Hence, T (and ) does not necessarly dagonalze A. Ths observaton s twofold. Frst, ths ponts out that the dervaton n [12] does not hold n general. Second, although optmal power allocaton for KM data streams seems very smlar to a general water fllng problem [10] wth multple water levels (one water level per each frequency band), ths allocaton cannot be determned by the algorthms n [10] [11]. Ths s because does not dagonalze A n (10), and hence we cannot convert (10) to a general water fllng problem. Pluggng (22) nto the Lagrangan functon (16), we have (17). The optmal power allocaton P () s obtaned by equatng the dervatve of (17) w.r.t P () to zero: L (,α,γ ) = P () D(α dag s (Π ) 1+P () dag s (Π ) dag s (Ω )=0 (23) Thus, ( ) P () =max 0, dag s (Π ) dag s (Ω ). (24) dag s (Π )dag s (Ω ) Pluggng (24) nto (17), we obtan the dual functon,γ ) n (18). To solve the DP (14) for α, k = 1,...,K,andγ (K +1 varables), we note that the problem s convex. Hence, any statonary pont s a globally optmal soluton. Moreover, as the obectve functon and constrants of the prmal problem (10) are contnuous w.r.t. every entry of, the dual functon D(α,γ ) s dfferentable w.r.t. and γ [26]. Hence, a gradent algorthm can be used α to obtan the optmal Lagrangan multplers α and γ by searchng for a statonary pont of the augmented Lagrangan of the DP n (19), where p s a postve penalty parameter (for volatng the constrants) and λ = {λ 1,...,λ K+1 } are nonnegatve multplers. Our gradent algorthm uses Armo step n the steepest descent drecton. Ths search mechansm together wth the above analyss are summarzed n Algorthm 1. By explotng the strong dualty, Algorthm 1 needs to deal wth only K +1 varables, nstead of 2KM 2 varables for the prmal problem (10). Before developng a centralzed algorthm that serves as a performance benchmark, we brefly dscuss how a MAC protocol can mplement Algorthm 1. D. MAC Protocol Usng ether a prened or frequency-hoppng-based control channel, we can desgn a MAC protocol that executes Algorthm 1. Ths protocol conssts of three wndows: Access wndow, tranng wndow, anddata wndow. The access wndow allows CR nodes to contend for channels. These nodes frst exchange RTS and CTS packets. Unlke IEEE , our MAC desgn does not use RTS/CTS packets to slence nearby nodes and reserve the transmsson floor for the upcomng transmssons. After ths phase, CR pars who have ust sent and receved RTS/CTS packets are admtted to the

6 2300 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 11, DECEMBER 2012 tranng wndow. The tranng wndow s used by nodes to exchange/negotate ther transmt strateges (precodng matrces). The sgnallng packets n ether access or transmt wndows can also be used to embed tranng sequences to obtan CSI matrces. The data wndow then follows wth multple data packets, sent usng negotated transmsson strateges. Ths approach s referred to as a flow-based approach n [14]. Algorthm 1 : Dstrbuted Algorthm for Power Allocaton and Spectrum Management 1: Input: = [ 1(t + 1),..., 1(t + 1), +1(t),..., N(t)] wth Gauss-Sedel teraton = [ 1(t),..., 1(t), +1(t),..., N(t)] wth Jacob teraton 2: Intalze (t +1) (t),γ 0; α 0, k Ψ K 3: whle true do 4: β.7,σ.1 (%used n Armo search) 5: λ k.1 k =1...(K +1) 6: p 1 7: whle L(α,γ,p,λ) 0do 8: step 0.1 9: D L(α,γ,p,λ) 10: d step D; m 0 11: (%fnd Armo step sze) 12: whle L(α σβ m step LD do 13: step step β; m m +1 14: d step D 15: end whle,γ,p,λ) L(α + d, γ + d, p, λ) 16: α α + d, γ γ + d 17: end whle 18: f mn(α,γ, k Ψ K) 0 break 19: k Ψ K : 20: λ k λ k pα 21: λ 1 λ 1 pγ f λ 1 pγ 0 else λ 1 =0 22: p p μ (%μ 1, ncrease cost of volaton) 23: end whle 24: Plug γ, α f λ k pα > 0 else λ k =0 nto (20) (Theorem 3) to fnd T P () k s found from (22) and (24). s found from (21). 25: RETURN (t +1), k Ψ K at tme (t +1) To reduce the overhead of the tranng wndow, one may relax the tme scale of recalculatng the prcng-factor matrx, tradng off throughput for less frequent updates. One can even omt the tranng wndow altogether by havng nodes embed updated nformaton nto every data and ACK packet. Then, upon recevng an ACK for each data packet, a transmtter recomputes ts prcng-factor matrx. Ths method s referred to as packet-based [28] [14]. An mportant ssue for protocol desgners s how to set the sze of the tranng wndow. Ths depends on the convergence speed of the updatng process. To ensure that the tranng wndow s not too long, the updatng and negotaton process must converge. Durng the tranng wndow, a node can use ether Gauss-Sedel (sequental) or Jacob (parallel) teratons (see Algorthm 1) to update ts precodng matrces. Although we cannot prove the convergence under the Jacob teraton, smulatons show that the dstrbuted algorthm converges even faster wth Jacob teratons than wth Gauss-Sedel. Convergence under the Gauss-Sedel teraton s clamed n the followng theorem. Theorem 4: Under the sequental updatng procedure (Gauss-Sedel), Algorthm 1 drves the game (10) to ts NE. Proof: See [25]. In [8] [11], the authors map the throughput maxmzaton game (of a sngle-frequency MIMO CRN) nto a varatonal nequalty problem (see [29] and theren references for a tutoral on varatonal nequalty theory) for the purpose of provdng the convergence and unqueness condtons of the NE. Intutvely, the game converges to a unque NE f there s not too much nterference at a recever and all channel matrces are full column-rank. The former condton requres transmtters to adust ther transmsson parameters and the latter s not always guaranteed. However, usng our proposed prcng polcy and the Gauss-Sedel procedure, the game (10) always converges to a NE wthout requrng transmtters and channel matrces to meet any addtonal requrements. Both Gauss-Sedel and Jacab updatng procedures are synchronous, requrng CR nodes to be n sync. Ths may not always be possble. By contrast, usng asynchronous update, some players, at some teratons, may not update other players wth ther strateges (e.g., due to packet collsons). Usng varatonal nequalty theory, the mean-value theorem, and followng the routne n [8], we can provde condtons under whch the game (10) converges to a unque NE under ether synchronous or asynchronous updates (we omt the detaled manpulaton due to space lmtaton). Theorem 5: The game (10) always converges to a unque NE f all channel matrces are full column-rank and the spectrum radus of the matrx J 1 Γ s less than 1, wherej s an N N dagonal matrx wth dagonal elements σ and Γ s a N N matrx of elements κ, wth the equaton at the top of the followng page. The operators eg mn (.) and eg max (.) gve the smallest and largest egenvalues of a matrx, respectvely. V. CENTRALIZED ALGORITHM A centralzed algorthm can be obtaned by formulatng the problem as a cooperatve game, where a network operator controls the behavors of all players n order to maxmze the network throughput. In ths secton, we use the augmented Lagrangan multpler method to derve such a centralzed algorthm. We rewrte the network-wde problem (6) as (25). The augmented Lagrangan [26] of (25) s gven n (26), where p s a postve penalty parameter (for volatng the constrants), and α and γ are nonnegatve Lagrangan multplers. At a local optmal soluton, (27) holds. The frst term n (27) s computed n (33). Its second term s gven by (30). Snce c and c k, are contnuously dfferentable w.r.t. every entry of, the thrd and fourth terms n (27) are also contnuously dfferentable [26]. Ther dervatves are as follows: {(max{0,γ + pc }) 2 { } 0 f γ + pc 0 = 2p(γ + pc ) {(max{0,α + pc k, }) 2 { } 0 f α }= + pc k, 0 2p(α + pc k, ) We use the gradent search algorthm wth Armo step sze [26] to fnd (,α,γ,p) such that (27) holds for all bands k and all users. The detals of the centralzed algorthm s presented n Algorthm 2. We emphasze that the network throughput may vary from a locally optmal pont to another. Hence, to account for such phenomenon, one can run the smulatons several tmes wth dfferent ntalzatons and take

7 NGUYEN and KRUNZ: PRICE-BASED JOINT BEAMFORMING AND SPECTRUM MANAGEMENT IN MULTI-ANTENNA COGNITIVE RADIO NETWORKS 2301 σ =mn{eg 2 mn H H k Ψ d(), I+ P mask (f k )H K Φ N ( ( ) ( κ, =max eg max H k Ψ d(), HH d(), eg max K Ψ N and κ, =0f = 1 d(), HH d(), ) H d(), H H d(), H d(), )) } mnmze R {,, Φ N } Φ N s.t. c = ) P max 0, Φ N c,k = ) P mask (f k ) 0, k Ψ K, Φ N (25) L(,α,γ,p)= R () + p {(max{0,γ +pc }) 2 (γ ) 2 }+ p 2 2 Φ N Φ N 0= = R L(,α,γ,p) Φ N \ x = [R[vec( ] [ )] T, I[vec( )] T T = [ x L =2 R[vec( L (1) 1 T )] R R[vec( (1),..., R[vec( Φ N + p 2 { {(max{0,γ + pc }) 2 } {(max{0,α + pc k, }) 2 (α ) 2 } (26) + {(max{0,α + pc k, }) 2 } } (27) ] )] T,..., R[vec( (K) )] T, I[vec( (1) )] T,..., I[vec( (K) )] T T L (K) N T )], I[vec( L (1) 1 T )],..., I[vec( L (K) N ] T T )] (28) (29) R =H H d(), (C d() +H d(), H H d(), ) 1 H d(), (30) the average throughput. The runnng tme for Algorthm 2 can be hgh as t nvolves NKM 2 complex varables (or 2NKM 2 real ones). To mplement Algorthm 2, we use the followng somorphsm mappng from a complex matrx to a vector of real varables. The vector of varables x =[(x T )N =1 ]T, wth x n (28) and the correspondng Lagrangan gradent n (29). VI. NUMERICAL RESULTS In ths secton, we evaluate the performance of the dstrbuted algorthm usng MATLAB smulatons. We compare the network throughput of the dstrbuted algorthm wth the centralzed one and the greedy algorthm, where nodes act selfshly to maxmze ther own rates. The greedy algorthm s exactly the same as the dstrbuted one except that the prcng-factor matrx A s a null matrx. Another algorthm called unform s obtaned by unformly dvdng a node s total transmt power over all avalable channels and then applyng the sngle-band approach n [9] for each channel. We emphasze that the unform algorthm nether meets the optmalty condtons (38) of the network problem (6) nor solves the per-user problem (10). Snce the number of varables n the centralzed algorthm s qute hgh (2NKM 2 ), ts runnng tme can be very long. To compare the four algorthms, we consder a CRN of 10 lnks, 3 channels (f 1 =2.4 GHz, f 2 =2.7 GHz, and f 3 =3GHz wth dentcal channel bandwdth of 1 MHz), and 4 antennas per node. The results are averaged over 30 runs. In each run, lnks are randomly placed n a square of length 100 meters. We take P max =2WandP mask =0.8 W for all channels. The channel fadng s flat wth attenuaton factor of 2. The spreadng angles of the sgnal at receved antennas vary from π/5 to π/5. For the lowest frequency, we assume that the receved power at a reference dstance of 100 meters reduces by 10 db compared wth the transmt power. To account for the frequency-dependent attenuaton factor, we assume that the receved power at the reference dstance decreases 2 db more f the frequency ncreases by 300 MHz. The PU nterference s treated as floor nose that together wth the thermal nose s normalzed to a unt varance. A snapshot of the network topology and antenna radaton patterns at steady state over channel f 2 s shown n Fg. 2. We can vsually notce that the transmtters under the dstrbuted and centralzed algorthms often steer ther beams away from neghborng recevers (three representatve pars of transmtter and a nearby unntended recever are hghlghted n ovals). Ths results from attemptng to mnmze the prce functon (11). It can also be seen that the antenna patterns

8 2302 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 11, DECEMBER 2012 Algorthm 2 : Centralzed Algorthm 1: Intalze I,γ 0; α 0, k Ψ K, Φ N 2: whle true do 3: β.7,σ.1 (%used n Armo search) 4: γ 0,α 0, k Ψ K, Φ N 5: p 1 6: whle L(,α,γ,p) 0do 7: step 0.1 8: D L(,α,γ,p) 9: d step D; m 0 10: (%fnd Armo step sze) 11: whle L(,α,γ,p) L( + d, α,γ,p) σβ m step LD do 12: step step β; m m +1 13: d step D 14: 15: end whle + d 16: end whle 17: f max(c,c,k, k Ψ K, Φ N ) 0 break 18: k Ψ K, Φ N : 19: γ = γ + pc f γ + pc 0 else γ =0 20: α = α + pc k, f α + pc k, 0 else α =0 21: p p μ (%μ 1, ncrease cost of volaton) 22: end whle 23: Return, k Ψ K, Φ N of the dstrbuted and centralzed algorthms are very smlar, suggestng the two algorthms may converge to the same pont. Fg. 3(a) depcts the network throughput under four algorthms (dstrbuted, centralzed, greedy, and unform) versus the number of teratons. Although the network performance at the converged ponts of the dstrbuted and centralzed algorthms change wth ther startng ponts, after averagng over multple runs wth dfferent ntalzatons, the throughput of the dstrbuted algorthm s almost the same as that of the centralzed one. We also notce that by usng the proposed prcng polcy to regulate nterference, the dstrbuted algorthm has almost twce the throughput of the greedy one. The unform algorthm also mproves network throughput over the greedy one but t remans nferor to our dstrbuted algorthm. Ths s because the unform algorthm evenly allocates power over all channels and does not optmze over the frequency dmenson, whle the dstrbuted algorthm attempts to optmze the antenna radaton patterns and power allocaton over both space and frequency. We say that an algorthm converges f the normalzed dfference n throughput between two consecutve teratons s less than a gven threshold (.e., 3%). The convergence speed of the dstrbuted algorthm versus the number of lnks s shown n Fg. 3(b). Fg. 3(c) depcts the network throughput under the dstrbuted and greedy algorthms versus the number of lnks. The dstrbuted algorthm consstently provdes hgher throughput than the greedy and unform algorthms. The mprovement becomes more sgnfcant wth more lnks. That s because as node densty ncreases (hgher number of lnks), network nterference ncreases, so nterference management becomes more crtcal n mprovng the throughput. To evaluate the energy effcency of the four algorthms, we record n Table I the average power consumpton and power allocaton per node for varous algorthms. Wthout regulatng nterference, nodes under the greedy algorthm selfshly compete for ther own throughput by always usng ther maxmum (a) Greedy algorthm (b) Dstrbuted algorthm (c) Centralzed algorthm Fg. 2. Antenna radaton patterns on channel 2 under the greedy, dstrbuted, and centralzed algorthms. power (2 W), leadng to the hghest power consumpton among the four algorthms. The power consumpton of the dstrbuted algorthm s comparable wth that of the centralzed and unform ones, and 10% less than that of the greedy one. Power allocaton over both space and frequency at a representatve node under the dstrbuted algorthm s shown Table II. From

9 NGUYEN and KRUNZ: PRICE-BASED JOINT BEAMFORMING AND SPECTRUM MANAGEMENT IN MULTI-ANTENNA COGNITIVE RADIO NETWORKS 2303 Network Throughput (Mbps) Iteratons to Converge Network Throughput (Mbps) Dstrbuted 40 Greedy 30 Centralzed Unform Iteratons (a) Jacob Gauss Sedel Number of Lnks (b) Dstrbuted Greedy Number of Lnks (c) Fg. 3. (a) Network throughput vs. teratons, (b) convergence speed of the dstrbuted algorthm, (c) network throughput vs. the number of lnks. Tables I and II, we notce that the nequalty constrants n problems (10) and (6) are not actve at ther solutons. That s because transmttng at hgh power may be expensve due to the proposed prcng method. VII. CONCLUSIONS AND FUTURE WORK In ths work, we nvestgated the spectrum sharng problem n a mult-antenna CRN. By adustng the precodng matrces, we optmzed the allocaton of power over both the frequency and space dmensons whle managng the antennas radaton beams to reduce network nterference, amng at maxmzng TABLE I AVERAGE POWER CONSUMPTION (IN WATTS) ALLOCATED OVER DIFFERENT CHANNELS. Channels Centralzed Greedy Dstrbuted Unform f f f Total TABLE II POWER ALLOCATION (IN WATTS) AT A NODE OVER SPACE AND FREQUENCY DIMENSIONS UNDER THE DISTRIBUTED ALGORITHM. Antennas f 1 f 2 f e e Total= network throughput. Usng game theory and the strong dualty n convex optmzaton, we desgned a low-complexty dstrbuted algorthm that acheves the same throughput as a locally optmal pont of the non-convex centralzed network problem. The key dea behnd the algorthm s the ntroducton of a dagonal block prcng-factor matrx for each CR. Ths matrx regulates network nterference by encouragng CRs to work n a cooperatve manner. Smulatons show that the proposed algorthm dramatcally mproves network throughput and acheves hgher energy effcency, compared wth exstng solutons. As a future work, one can extend the proposed prcng polcy to coordnated mult-cell systems and also heterogeneous spectrum sharng networks n whch the sets of avalable frequences at nodes are dfferent. Moreover, because CSI s vulnerable to estmaton errors, one may wsh to desgn a robust game model to deal wth partal CSI. APPENDIX A PROOF OF THEOREM 2 The acheved NE s characterzed by the solutons of all N per-user optmzaton problems (10). Snce the ndvdual utlty optmzaton problem s convex, a locally optmal soluton s globally optmal. The optmal soluton can be found by solvng ts K.K.T. condtons [26], gven by: L (,α,γ ) = R A (α +γ ) =0, k ) P mask (f k ) 0, k Ψ K ) P mask ]=0, k Ψ K α [ ) P max 0 γ [ ) P max ]=0 (31) The Lagrangan functon of the network optmzaton problem (6) s n (32), where =, the set of precodng matrces over all users and frequency bands. All the statonary (or locally optmal) ponts of the network

10 2304 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 11, DECEMBER 2012 L(,α,γ )= Φ N R L (,α,γ )= R Φ N α [ = d(), C d() H d(), + H H d(), (C d() + H d(), = d(), C d() H d(), {log det(i+ L (,α,γ )= {α P mask (f k )+ γ L = { tr( T H L { tr( T H [( d(), C d() H d(), K P max tr( T H T T )+ ) P mask (f k )] γ [ Φ N H H d(), ) 1 H d(), ) P max ] (32) ) 1 + d(), C d() H d(), ] 1 d(), C d() H d(), T )+log det(i+ T H M s=1 log(1+dag s { T H ) tr( [A +(α )+log det(i+ T H +γ )I] E 1 d(), C d() H (33) )+α P mask (f k )+ γ K P max} d(), EH 1 (34) T )} (35) E 1 H 1 H d(), C 1 d() H d(), EH T )} (36) E 1 H 1 H d(), C d() H d(), EH 1 T })} (37) problem must satsfy ts K.K.T. condtons: L(,α,γ ) α [ γ [ = R + {Φ N \} R (α +γ ) =0 ) P mask (f k ) 0, k Ψ K, Φ N ) P mask ]=0, k Ψ K, Φ N ) P max 0, Φ N ) P max ]=0, Φ N (38) To guarantee that the game (10) wth the prce functon ned n (11) converges to a NE at whch the CRN s throughput s the same as that of a locally optmal soluton to problem (6), the NE of the game (10) must be a statonary pont of problem (6). In other words, the K.K.T. condtons of (6) have to hold at the statonary pont of (10). For that to happen, the followng equalty must hold (through comparng condtons (31) and (38)): To compute R C d() = I+H + A = {Φ N \} R. (39) n (33), recall (2) and note that: d(), H d(),v v v Φ N \{,} H H d(), v H H H d(),v. The last equalty n (33) follows by applyng the Woodbury dentty [30] to (C d() +H d(), H H d(), ) 1. Pluggng (33) nto (39), we get (13). One can also easly verfy that the derved A matrx s postve-semnte. Addtonally, f the prcng-factor has the form (13), the acheved NE meets the K.K.T. condtons of the network-wde problem (6). APPENDIX B PROOF OF THEOREM 3 Let s rewrte the Lagrangan functon of (10) as n (34), then usng the Cholesky decomposton [A +(α + γ )I] = E E H. We have (35), where T H = E. As P max and P mask (f k ) are predetermned values and from (35), L s maxmzed f we maxmze L n (36). Followng the routne of usng Hadamard nequalty (e.g., [9], [19]), applyng the Hadamard s nequalty [31] to the second term of (36), we have (37), where dag s (.) s the (s, s) dagonal element of a matrx (.). The equalty happens when T H E 1 H 1 H d(), C 1 d() H d(), EH T s a dagonal matrx. Ths s the case f there exsts an orthonormal matrx T that dagonalzes the matrx 1 1 H. Hence, we should have: E 1 H H d(), C d() T H T T 1 E = I d(), EH 1 H 1 H d(), C 1 d() H d(), EH T = Λ (40) where Λ s a M M dagonal matrx. Multplyng both sdes of (40) by T,thenE,wehave: d(), C 1 d() H d(), EH T Recall that T H = decomposton we have: d(), C d() H d(), Ths concludes the proof. E = E =[A = E T Λ. and the above Cholesky E H REFERENCES +(α Λ + γ )I] Λ. [1] D.TseandP.Vswanath,Fundamentals of Wreless Communcaton. Cambrdge Unversty Press, May 2005.

11 NGUYEN and KRUNZ: PRICE-BASED JOINT BEAMFORMING AND SPECTRUM MANAGEMENT IN MULTI-ANTENNA COGNITIVE RADIO NETWORKS 2305 [2] K. Sundaresan, R. Svakumar, M. Ingram, and T.-Y. Chang, A far medum access control protocol for ad-hoc networks wth MIMO lnks, n Proceedngs of the INFOCOM Conference, vol. 4, March 2004, pp [3] S. Chu and X. Wang, Opportunstc and cooperatve spatal multplexng n MIMO ad hoc networks, n Proceedngs of the Mobhoc Conference, 2008, pp [4] G. Scutar and D. Palomar, MIMO cogntve rado: A game theoretcal approach, IEEE Transactons on Sgnal Processng, vol. 58, no. 2, pp , Feb [5] Z.-Q. Luo and S. Zhang, Dynamc spectrum management: Complexty and dualty, IEEE Journal of Selected Topcs n Sgnal Processng, vol. 2, no. 1, pp , Feb [6] Y. J. Zhang and A. So, Optmal spectrum sharng n MIMO cogntve rado networks va semnte programmng, IEEE Journal on Selected Areas n Communcatons, vol. 29, no. 2, pp , Feb [7] R. Zhang and Y.-C. Lang, Explotng mult-antennas for opportunstc spectrum sharng n cogntve rado networks, IEEE Journal of Selected Topcs n Sgnal Processng, vol. 2, no. 1, pp , Feb [8] J. Wang, G. Scutar, and D. Palomar, Robust MIMO cogntve rado va game theory, IEEE Transactons on Sgnal Processng, vol. 59, no. 3, pp , March [9] S.-J. Km and G. B. Gannaks, Optmal resource allocaton for MIMO ad hoc cogntve rado networks, IEEE Transactons on Informaton Theory, vol. 57, no. 5, pp , May [10] D. Palomar and J. Fonollosa, Practcal algorthms for a famly of waterfllng solutons, IEEE Transactons on Sgnal Processng, vol. 53, no. 2, pp , Feb [11] G. Scutar, D. Palomar, and S. Barbarossa, Asynchronous teratve water-fllng for gaussan frequency-selectve nterference channels, IEEE Transactons on Informaton Theory, vol. 54, no. 7, pp , July [12] X. Dong, Y. Rong, and Y. Hua, Cooperatve power schedulng for a network of MIMO lnks, IEEE Transactons on Wreless Communcatons, vol. 9, no. 3, pp , March [13] G. Scutar, D. Palomar, and S. Barbarossa, Compettve desgn of multuser MIMO systems based on game theory: A unfed vew, IEEE Journal on Selected Areas n Communcatons, vol. 26, no. 7, pp , Sept [14] F. Wang, M. Krunz, and S. Cu, Prce-based spectrum management n cogntve rado networks, IEEE Journal of Selected Topcs n Sgnal Processng,, vol. 2, no. 1, pp , [15] J. Hrshlefer, A. Glazer, and D. Hrshlefer, Prce Theory and Applcatons Decsons, Markets, and Informaton. Cambrdge Unversty Press, [16] M. J. Osborne, An Introducton to Game Theory. Oxford Unversty Press, [17] R. Ilts, S.-J. Km, and D. Hoang, Noncooperatve teratve MMSE beamformng algorthms for ad hoc networks, IEEE Transactons on Communcatons, vol. 54, no. 4, pp , [18] C. Sh, R. Berry, and M. Hong, Local nterference prcng for dstrbuted beamformng n MIMO networks, n Proceedngs of the IEEE MILCOM Conference, 2009, pp [19] D. Hoang and R. Ilts, Noncooperatve egencodng for MIMO ad hoc networks, IEEE Transactons on Sgnal Processng, vol. 56, no. 2, pp , [20] FCC, Spectrum polcy task force report, ET Docket No and No , Sep [21] W. Yu, Competton and cooperaton n mult-user communcaton envronments, PhD Thess, Stanford Unversty, Standford, CA, [22] W. Wang, Y. Cu, T. Peng, and W. Wang, Noncooperatve power control game wth exponental prcng for cogntve rado network, n Proceedngs of the IEEE 65th Vehcular Technology Conference, Aprl [23] G. Arslan, M. Demrkol, and S. Yuksel, Power games n MIMO nterference systems, n Proceedngs of the Internatonal Conference on Game Theory for Networks, May 2009, pp [24] J.-S. Pang, G. Scutar, D. Palomar, and F. Facchne, Desgn of cogntve rado systems under temperature-nterference constrants: A varatonal nequalty approach, IEEE Transactons on Sgnal Processng, vol. 58, no. 6, pp , June [25] D. Nguyen and M. Krunz, Spectrum management and power allocaton n MIMO cogntve networks, Unversty of Arzona, Tech. Rep. TR-UA-ECE , August [Onlne]. Avalable: krunz [26] D. P. Bertsekas, Nonlnear Programmng. Athena Scentfc, [27] R. A. Horn and C. R. Johnson, Matrx Analyss. Cambrdge Unversty Press, [28] R. Maheswaran and T. Basar, Decentralzed network resource allocaton as a repeated noncooperatve market game, n Proceedngs of the 40th IEEE Conference on Decson and Control, vol. 5, 2001, pp [29] G. Scutar, D. Palomar, F. Facchne, and J.-S. Pang, Convex optmzaton, game theory, and varatonal nequalty theory, IEEE Sgnal Processng Magazne, pp , May [30] W. W. Hager, Updatng the nverse of a matrx, SIAM Revew. A Publcaton of the Socety for Industral and Appled Mathematcs, vol. 31, no. 2, pp , [31] A. W. Marshall and I. Olkn, Inequaltes: Theory of Maorzaton and Its Applcatons. Academc Press, Dep N. Nguyen receved the B.S. degree n telecommuncatons and electroncs from Posts and Telecommuncatons Insttute of Technology (PTIT), Vetnam, n 2004, and the M.E. degree n electrcal and computer engneerng from the Unversty of Calforna, San Dego n From 2004 to 2006, he was a lecturer at the Department of Telecommuncatons, PTIT, Vetnam. He was a member of techncal staff at Broadcom Corporaton from 2008 to He s currently workng towards the Ph.D. degree at the Department of Electrcal and Computer Engneerng, The Unversty of Arzona. Hs research nterests nclude cogntve rados, ad hoc, and sensor networks, wth emphass on resource allocaton and network management for mult-antenna systems. Marwan Krunz receved the PhD degree n electrcal engneerng from Mchgan State Unversty n He has been wth the Unversty of Arzona snce 1997, where he s a professor n the Electrcal and Computer Engneerng and Computer Scence Departments. He s the UA ste drector for Connecton One. He has held several vstng research postons and was a Char of Excellence at the Unversty of Carlos III, Madrd, Span. Hs research nterests nclude computer networkng and wreless communcatons wth a focus on dstrbuted rado resource management n wreless and sensor networks, protocol desgn, and secure communcatons. He has publshed more than 170 ournal artcles and refereed conference papers and s a conventor on three US patents. He was a recpent of the US Natonal Scence Foundaton (NSF) CAREER Award n He has served on the edtoral boards for the IEEE Transactons on Network and Servce Management, IEEE/ACM Transactons on Networkng, IEEE Transactons on Moble Computng, and Computer Communcatons Journal, and as a TPC char for several nternatonal conferences. He s a fellow of the IEEE.