Spectrum Management and Power Allocation in. MIMO Cognitive Networks

Transcription

1 Spectrum Management and Power Allocaton n 1 MIMO Cogntve Networks Dep N. Nguyen and Marwan Krunz Department of Electrcal and Computer Engneerng, Unversty of Arzona E-mal:{dnnguyen, krunz}@ece.arzona.edu Techncal Report TR-UA-ECE Last Updated: January 16, 2012 Abstract Cogntve rado (CR) technques mprove the spectrum utlzaton by explotng temporarly-free frequency bands (.e., n the tme doman). The spectrum utlzaton can be boosted further f CR nodes are equpped wth multple antennas to leverage communcatons n the spatal dmenson. In ths artcle, we consder the problem of maxmzng the throughput of a mult-nput mult-output (MIMO) cogntve rado network. Wth spatal multplexng performed over each frequency band, a mult-antenna CR node controls ts antenna radaton pattern and allocates power for each data stream by approprately adjustng ts precodng matrx. Our objectve s to desgn a set of precodng matrces (one for each band) at each CR node so that power and spectrum are optmally allocated for that node (n terms of throughput) and ts nterference s steered away from other CR and PR transmssons. In other words, the problems of power, spectrum allocaton and nterference management are jontly nvestgated. We come up wth a mult-carrer MIMO network throughput optmzaton problem subject to frequency-dependent power constrants. The problem s non-convex, wth the number of varables growng quadratcally wth the number of antenna elements. To tackle t, we translate t nto a noncooperatve game. We then 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

2 ths NE s equal 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), a low-complexty dstrbuted algorthm s developed by explotng the strong dualty of the per-user convex optmzaton problem. The number of varables n the dstrbuted algorthm s ndependent to the number of antenna elements. A centralzed (cooperatve) algorthm s also developed, servng as a performance benchmark. Smulatons show that the network throughput under the dstrbuted algorthm converges rapdly to that of the centralzed one. We then develop a MAC protocol that mplements our resource allocaton and beamformng scheme. Extensve smulatons show that the proposed protocol dramatcally mproves the network throughput as well as reduces the power consumpton. The applcaton of our results s not lmted to CR systems, but extends to any mult-carrer (e.g., OFDM) MIMO system. Index Terms Noncooperatve game, prcng, cogntve rado, MIMO, MAC protocol, power allocaton, frequency management, beamformng. I. INTRODUCTION Recent years have wtnessed great research nterests n cogntve rado (CR) and mult-nput multoutput (MIMO) systems. Whle the former s vewed as a key enablng technology to mprove spectrum utlzaton, the later has already proved tself as a powerful sgnal processng technque to mprove spectral effcency. Through sensng and/or probng, CRs can opportunstcally communcate on temporarly avalable spectrum bands whle avodng nterference wth lcensed-spectrum (or prmary rado-pr) users. MIMO communcatons mprove the channel capacty by sendng ndependent data streams smultaneously over dfferent antennas (a technque known as spatal multplexng). A crucal challenge n CR research s how to effectvely allocate transmsson powers and spectrum among CRs (see Fgure 2(a)) so as to maxmze network throughput whle avodng nterferng wth PR receptons. Even for a sngle frequency band and sngle-antenna wreless devces, the problem s dffcult due to the non-convexty of the network throughput functon. For sngle-antenna CRs, dstrbuted algorthms were developed n [1] [2] usng game theory. The ncorporaton of MIMO technques nto CR systems ntroduces two new control dmensons (besdes 2

3 Fg. 1. MIMO precodng method. power control and frequency management): power allocaton over antennas (space dmenson) and nterference management. The latter comes from MIMO s degrees of freedom [3], whch allow a MIMO node to suppress nterference from others (by usng some of ts antennas) and confgure ts antenna radaton patterns to keep nterference away from unntended recevers. MIMO s power allocaton and nterference management can be jontly controlled va precodng matrces, a spatal multplexng technque [3] (Fgure 1). Usng ths technque, the vector of nformaton symbols are pre-multpled wth a matrx before beng placed on a transmt antenna array. By tunng the ampltude and the phase of each complex entry n the precodng matrx, one adjusts not only the allocated powers but also the radaton drectons, whch together shape the antenna radaton patterns. Prevous MIMO-networkng works [4] [5] [6] consdered power allocaton or stream control (Fgure 2(b)) but dd not take nto account nterference management va controllng antenna beams. An optmal set of precodng matrces for each node would be one that allocates power over both space and frequency dmensons (2(c)) and yelds radaton patterns that nduce mnmum nterference (2(d)), so as to maxmze the network throughput. Ths problem s the focus of our work. II. RELATED WORKS If one gnores the need to protect PR receptons, a MIMO-based CR system very much resembles a multcarrer (e.g., OFDM) MIMO (MC-MIMO) system. In MC-MIMO, jont power and spectrum optmzaton s a non-convex problem, whch s attrbuted to co-channel mult-user nterference. Globally optmal solvers for non-convex problems, e.g., branch and bound, often have exponentally growng complexty n the number of varables. Unfortunately, the number of varables n a MC-MIMO network can be very large. For nstance, when usng the precodng technque wth 4 antennas per node and 10 sub-carrers n a network of 10 lnks, 3

4 (a) (b) (c) (d) Fg. 2. Power allocaton n ether frequency (a) or space dmenson (b) or n both dmensons (c) and four transmt radaton patterns steerng away from nearby recevers (d). ths number s = 1600 complex varables (or3200 real varables). The latest advances n power and spectrum management for MC-MIMO can be found n [7] usng dual stochastc optmzaton. Smlar to the aforementoned works, the authors n [7] [8] only consdered the power allocaton (the ampltude) but dd not optmze the antenna radaton drectons (the phase). In addton to mposng constrants to protect PRs, we consder a full/generalzed egen precodng MIMO CR network. In the lterature, there have been a vast body of works on MIMO precodng-matrx desgn, categorzed nto beamformng and generalzed egencodng. In beamformng (e.g., [9] [10] for MIMO networks and [11] for MIMO CRNs), there s only one data stream to be sent, hence all precodng matrces reduce to vectors (matrces of rank one). In generalzed egencodng, there s no constrant on the rank of the precodng matrces [12],.e., several data streams can be sent smultaneously. Inspred by the ntroducton of the spatal multplexng technque nto exstng networks (e.g., n allows up to four concurrent multplexed streams), generalzed egencodng has recently attracted great nterests. There have been recent works at the physcal layer that attempt to protect PR communcatons n a MIMO CR network (CRN) whle maxmzng the CRN s throughput (e.g., [11] [13] [14]). These works assumed full or partal avalablty of channel state nformaton (CSI) from each CR to each PR. Ths requres feedback or coordnaton between CRs and PRs. However, current lcensed rado devces are not ready for 4

5 such a feedback mechansm, as CR communcatons are expected to be transparent to PRs. Authors n [15] provded a MIMO CR scheme whch s robust to CR-PR CSI estmaton error by dealng wth the worst case error. In our setup, PR communcatons are protected by mposng frequency-dependent constrant on the transmsson power of CRs. Ths assumpton has been used n sngle-antenna CRNs e.g., [2] [16]. Our work s n lne wth the jont resource allocaton and waveform adaptaton developed n [16] for sngle antenna CRNs and wthout a prcng polcy. Though we lmt our analyss to the case of usng power masks to focus on the beamformng and spectrum sharng among MIMO CR nodes, t should be noted that the formulaton and analyss here are also applcable to CR systems that adapt the CR transmsson parameters to the surroundng envronment [17] as well as the error-robust scheme n [15]. Because of the challenges assocated wth power and spectrum optmzaton, most exstng works on MIMO CR systems (e.g., [18] [15] [14]) do not consder optmzaton over the frequency dmenson. The extenson of these works to mult-band MIMO CRNs s not trval. Frst, scalar-value algorthms (e.g., the bsecton search n [14]) used for a sngle-band MIMO ad hoc network wll not work when searchng for optmal vectors n mult-band MIMO CRNs. Second, we show that even wthout beamformng, the optmal power allocaton over both frequency and spatal dmensons s not equvalent to a general water fllng problem [19](wth multple water levels), then cannot be treated usng exstng algorthms [20] [19] [21]. Thrd, f one separately apples results from sngle-band MIMO to each ndvdual band of a mult-band MIMO CR system, the acheved throughput s often low, as the resultng network operaton ponts do not meet the optmal condtons (dscussed later). In ths paper, smulatons are used to compare the performance of such approaches to that of our algorthms. Addtonally, t should be noted that the MIMO CR game-based analyss n [18] [15] dd not use a prcng mechansm. Smulatons showed that our prcng technque greatly outperforms non-prcng-based mechansms n terms of both network throughput and power effcency. Motvated by the above, the objectve of ths paper s to develop low-complexty dstrbuted algorthms that confgure the transmt antenna radaton drectons and allocate power over all data streams, specfed by space (subndex s) and frequency (subndex k) dmensons, so that the MIMO CRN s throughput s maxmzed. We model ths jont problem of power, spectrum allocaton and beamformng as a prce-based 5

6 noncooperatve game [22]. To manage nterference, we derve a dagonal block prcng-factor matrx. Ths matrx s user-dependent, and s used to capture the nterference effect from a transmtter to unntended recevers. Hence, t s a functon of the node s precodng matrces as well as ts neghborng recevers. The prcng-factor matrx does not only mprove the Nash-Equlbrum (NE) of the game, but also drves the game towards a locally optmal pont of the centralzed problem. Explotng the strong dualty n convex optmzaton, we desgn a low-complexty dstrbuted algorthm to determne the set of precodng matrces (best reponse) for each node. The dmensonalty of the dstrbuted algorthm s only K+1, where K s the number of frequency bands,.e., t does not depend on 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 results show that the performance of the dstrbuted algorthm s almost the same as that of the centralzed one. Throughout the paper, we use (.) to denote the conjugate 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 ndcated n boldface. 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 usng augmented Lagrangan multplers n Secton V. Numercal results are dscussed n Secton VI. Concludng remarks are provded n Secton VII. III. PROBLEM FORMULATION We consder a CRN that coexsts wth several PR networks. The CRN conssts of N lnks. Each CR node s equpped wth M antennas. The spectrum to be allocated s comprsed of K orthogonal frequency def bands (referred to as channels) that have central frequences f 1, f 2,..., f K. Let Φ N = {1,2,...,N} def and Ψ K = {1,2,...,K} denote the sets of CR lnks and channels, respectvely. Each CR user can smultaneously communcate over multple frequency bands, denoted by the set S. We mpose a half- 6

7 duplex constrant on all transmssons, meanng that a CR cannot transmt and receve at the same tme. The transmtter of each CR lnk can send up to M ndependent data streams on ts M antennas over a gven channel. A node controls the emtted antenna pattern and power allocaton for these streams through ts precodng matrces. Formally, for frequency band f k, let x (k) be a column vector of M nformaton symbols, sent from node to ts destnaton node d(). Each element of x (k) belongs to one data stream. Let T (k) denote the precodng matrx of node on frequency band f k. Then, the actual transmt vector s T (k) x (k). We allow for spectrum sharng among dfferent CR lnks. Ths assumpton s n contrast to the case that restrcts to one CR transmsson on a gven frequency channel n the same neghborhood. In our setup, several CR lnks can smultaneously occupy the same channel. Specfcally, for a frequency band f k, the receved sgnal vector y (k) d() at the recever d() of lnk (,d()) s gven by: y (k) d() = H(k) d(), T (k) x (k) + j Φ N \{} H (k) d(),j T (k) j x (k) j +N k (1) where the frst term n the RHS of (1) s the desred sgnal sent from transmtter and H (k) d(), s the channel gan matrx on frequency band f k from the transmtter to the recever d(). Specfcally, H (k) d(), h 1 h 2...h M, where h s s an M 1 column vector of channel gans from transmt antenna s to all M recevng antennas, s = 1,...,M. We assume a flat-fadng channel. Each entry of H (k) d(), def = s a complex Gaussan varable wth zero mean and unt varance. The second term n the expresson of y (k) d() represents nterference from other CR lnks that share channel f k wth lnk (,d()). N k s an M 1 complex Gaussan nose vector wth dentty covarance matrxi, representng the nose floor as well as normalzed (and whten) nterference from nearby PR users on band k. The Shannon capacty of lnk (,d()), referred to as () for short, on the frequency band f k s [3]: () = logdet(i+ H (k)h 1 d(), C(k) (k) d() H d(), T (k) ) (2) R (k) 7

8 where C (k) d() s the nose-plus-nterference covarance matrx at d() over frequency band f k, gven by: C (k) d() = I+ H (k) d(),j T (k) j j Φ N \{} j H (k)h d(),j. The total channel rate over all frequency bands of lnk s: We use P () s,k R () = R (k) (). (3) to denote the power allocated on band k (frequency dmenson) at stream s (space dmenson or antenna) of CR user. P () s,k s the entry (s,s) on the dagonal of matrx ( T (k) ). For user, the total power allocated on all frequency bands and all antennas should not exceed ts maxmum power budget P max (we assume an dentcal power lmt for all CR users). Consequently, M P () s,k = tr( T (k) ) P max. (4) s=1 Spectrum sharng between CR and PR transmssons takes two forms: spectrum overlay and spectrum underlay. In the former, CRs only occupy a channel f on that channel, no PR s detected (also known as a detect-and-avod mechansm). Spectrum underlay allows CR users to occupy a channel even f PRs are detected, provded that the transmssons of CRs do not deterorate the qualty of servce for the PR users. There are two methods to realze underlay spectrum sharng: statc and dynamc. The statc method requres that the transmt power of CRs on frequency channel f k s always less than a gven power mask def P mask (f k ). Let P mask = (P mask (f 1 ),P mask (f 2 ),...,P mask (f K )) denote the power mask on all channels. Instead of specfyng hard constrants on the transmt powers of CRs, the dynamc method adapts these transmt powers to actvtes from neghborng PRs and other CRs so that the accumulated nterference (from all CRs and PRs) at a nearby PR recever does not exceed a threshold. Though the dynamc method may result n hgher network throughput, t requres coordnaton among CRs and PRs and can only statstcally guarantee the PRs qualty of servce. Ths s due to the fact that t s mpractcal to perfectly model and 8

9 estmate the nterference from CR and PR lnks. In ths paper, we use the statc method, mplyng that: M s=1 P () s,k (k) = tr( T ) P mask (f k ). (5) It should be noted that the subsequent analyss s also applcable to the dynamc method and the detectand-avod mechansm. We am at maxmzng the CRN throughput. Mathematcally, the network optmzaton problem can be stated as follows: s.t. C1: maxmze { T (k),, Φ N } Φ N R () tr( T (k) ) P max, Φ N C2: tr( T (k) ) P mask (f k ), k Ψ K, Φ N. (6) IV. GAME THEORETIC DESIGN The network optmzaton problem (6) s not convex due to the presence of nterference among CR users that share the same frequency band. Thus, even n a centralzed manner, computng the globally optmal soluton s prohbtvely expensve. Thus, we reformulate t usng game theory and derve a prcng functon for each CR lnk that guarantees a locally optmal soluton for problem (6), found n a dstrbuted manner. A. Game Formulaton A noncooperatve game s characterzed by ts set of players, ther acton/strategy space, and ther utlty/payoff functons. For the underlyng 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, subject 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 T def = ( T {1}, T {2},..., T (k) ), whch can be vewed as an M KM block matrx, comprsed of K M M def matrces. Let T = ( T 1, T 2,..., T 1, T +1,..., T N ) be the set of actons from all lnks, except lnk. 9

10 The utlty or payoff of player for ts acton T s mapped to lnk s Shannon rate, whch also depends on the selecton of the precodng matrces from other CR lnks T : U ( T, T ) = def R () = logdet(i+ H (k)h 1 d(), C(k) (k) d() H d(), T (k) ). (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 drectons to maxmze ts own return. Formally, each CR user solves the followng problem for ts precodng matrx set T : maxmze { T (k), } U ( T, T ) s.t. C1 : C2 : tr( T (k) ) P max tr( T (k) ) P mask (f k ), k Ψ K. (8) By solvng the above problem, CR users mplctly nteract wth each other through ther choce of the precodng matrces. Under some condtons, the game reaches a NE where no CR user has ncentve to unlaterally devate from. However, as each CR user behaves selfshly, the resultng NE s often far from the Pareto optmum, and network throughput can be low. To drve the above noncooperatve game to a better NE,.e., acheve hgher socal welfare, we use a prcng or taxaton mechansm to encourage selfsh players to work n a cooperatve manner [23]. Prcng makes players more socally responsble for ther actons. The utlty functon wth prce s defned as follows: U ( T, T ) = def U ( T, T ) F u ( T ) (9) where F u ( T ) s the prcng functon for lnk. Consequently, we come up wth the followng noncooperatve 10

11 game wth prcng: maxmze { T (k) } s.t. U ( T, T ), Φ N (10) C1 and C2 as n problem (8). B. Prcng Polcy A Pareto-optmal prcng polcy s one that drves the game to a NE on the Pareto fronter. An optmal prcng polcy s one that yelds the game to a NE that s dentcal to the globally optmal soluton of the non-convex problem (6). However, dervng such a prcng functon s often dffcult for two reasons. Frst, t s hard to characterze the optmal or the Pareto-optmal prcng polcy, makng t not possble to quantfy the performance gap between these polces and the acheved NE. Second, an optmal prcng functon that requres global network nformaton s mpractcal for a dstrbuted network. To mprove the effcency of the NE, the prcng functons n the lterature are usually based on heurstcs. For nstance, the prcng functons n [24] are suboptmal lnear functons wth a fxed prcng-factor. In economcs, the prcng functon can take varous forms to account for varous marketng and prcng polces, e.g., volume dscount, coupon dscount, etc. In the context of network resource allocaton, both lnear (e.g., [2] [25]) and nonlnear [1] prcng functons have been proposed. In ths paper, we defne the prcng functon F u ( T ) as follows: ] F u ( T ) = tr [ T H A T (11) where A = A (1) A (2) A (K) (12) s an KM KM block dagonal matrx, consstng of K blocks along ts dagonal. The kth block A (k) s 11

12 an M M postve-semdefnte matrx. A s referred to as the prcng-factor matrx of CR lnk and A (k) s referred to as the prcng-factor matrx at frequency band 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 Appendx I. The above game can have more than one NE. To guarantee a lower bound for the effcency on the acheved NE, we propose n the next theorem 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: For the game n (10) to converge to a NE at whch the CRN s throughput equals to that of a locally optmal soluton of problem (6), the prcng-factor matrx A n (12) must have ts k block matrx A (k) of the followng form: A (k) = j Φ N \{} Proof: See Appendx II. H (k)h 1 d(j), C(k) (k) d(j) H d(j),j [( T (k) j j ) 1 +H (k)h 1 d(j),j C(k) (k) d(j) H d(j),j ] 1 H (k)h 1 d(j),j C(k) (k) d(j) H d(j), (13) To compute the prcng-factor matrx A n (12), a CR transmtter needs to obtan feedback regardng the nterference-plus-nose covarance, the precodng, and the channel matrces from all other lnks. In practce, f the channel gan matrx from to d(j) s too weak,.e., H (k) d(j), 0, there s no need for d(j) to send ts feedback to. Hence, only gets feedback from recevers d(j) that are wthn s vcnty. It s also worth notng that the feedback nformaton s locally avalable at a recever d(j) as a byproduct of ts decodng process (.e., successve nterference cancelaton (SIC) recevers [3]). The kth block A (k) of the prcng factor matrx n (13) agrees wth that has been derved n [14] for a sngle-band MIMO CRN usng frst order Taylor seres approxmaton. That s because we assume that all bands are orthogonal. However, n our case, the prcng matrx s a block dagonal matrx. Addtonally, the necessary condtons of A (k) have not been derved n the sngle-band case. How a MAC protocol desgn can support the computaton of the above prcng-factor matrx at a node wll be dscussed shortly. 12

13 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 the actons of other CR lnks. Notng that problem (10) s convex, hence can be solved by standard methods, e.g., nteror pont [26], requrng polynomal tme w.r.t to the problem s number of varables. Authors n [11] solved a smlar problem usng semdefnte programmng. However, as mentoned before, the number of varables of (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 to the antenna array sze. It should also be noted that usng our proposed prcng polcy, the per-user optmzaton problem cannot be solved by water fllng algorthms n [19] [20] as the optmal precodng matrx does not necessarly dagonalze the prcng-factor matrx n (10). Recallng the convexty of (10) and that the Slater s condtons can easly be shown to hold [27], strong dualty holds for problem (10),.e., an optmal soluton T to (10) should also solve the followng dual problem (as n the case of a sngle-channel MIMO network [14]): DP : mnmze {α (k),γ 0, } where D(α (k),γ ) s the dual functon, defned as: D(α (k),γ ) (14) D(α (k),γ ) = maxmze { T (k), } L ( T,α (k),γ ). (15) wth L ( T,α (k),γ ) s the Lagrangan functon defned n (30). The optmal matrx T of (10) s featured n the followng theorem. Theorem 3: The M KM block matrx T that solves the ndvdual utlty optmzaton problem (or the user s best response) must have ts kth block, the matrx T (k), n a form of the generalzed egen matrx of the matrx H (k)h 1 d(), C(k) (k) d() H d(), and matrx A(k) +(α (k) +γ )I, where α (k) and γ are the optmal Lagrange 13

14 multplers of (10). In other words, the followng equatons must hold k Ψ K : where Λ (k) H (k)h 1 d(), C(k) (k) d() H d(), T (k) = [A (k) +(α (k) +γ )I] T (k) Λ (k) s a gven M M dagonal matrx. (16) Proof: See Appendx III. As prevously dscussed, the precodng matrx T (k) determnes both the drectons of the antenna radaton as well as how node allocates ts transmsson power on dfferent antennas over frequency band k. Theorem (3) states a class of matrces that the soluton of (10) must belong to. Ths class provdes the drectons that a user should pont ts antenna radaton to. The next step s to fnd the optmal power allocaton P () s,k for the set of KM data streams. To ensure that T (k) belongs to the class of matrces specfed by Theorem (3), we let: T (k) = T (k) P () 1/2 k (17) where T (k) s an M M matrx wth unt-norm column vectors that satsfes (16). Ths matrx can be found by normalzng the generalzed egen matrx T (k). 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 to sub-channel (s,k), P () s,k. We can verfy that the expresson of T (k) n (17) satsfes (16). As T (k) s a generalzed egen matrx of matrces H (k)h 1 d(), C(k) (k) d() H d(), and A(k) +(α (k) +γ )I, then T (k) must also dagonalze each of the two matrces [28]: T (k)h [H (k)h 1 d(), C(k) (k) d() H d(), ]T(k) = Π (k) (18) T (k)h [A (k) +(α (k) +γ )I]T (k) = Ω (k) (19) where Π (k) and Ω (k) are M M dagonal matrces. Note that though ts columns have unt-norm, T (k) s not an orthonormal matrx as A (k) s generally not smlar to I. Hence, T (k) (then T (k) ) does not dagonalze A (k). Ths observaton s twofold. Frst, ths 14

15 ponts out that the the soluton to the problem n [21] was ncorrectly derved. Second, though the optmal power allocaton over KM data streams problem seems very smlar to a general water fllng problem [19] wth multple water levels (one water level per each frequency band), t cannot be solved by the algorthm developed n [19]. Ths s because T (k) a general water fllng problem. does not dagonalze A (k) n (10), hence we cannot convert (10) to Pluggng (18) and (19) nto the Lagrangan functon (30), we have: L ( T,α (k),γ ) = { M s=1 {log(1+p () s,k dag s (Πk )) P () s,k dag s (Ω(k) )}+α (k) P mask (f k )+ γ K P max}. (20) The optmal power allocaton P () s,k to zero: s obtaned by equatng the dervatve of the Lagrangan (20) w.r.t P() s,k L ( T,α (k),γ ) P () s,k = dag s (Π k ) 1+P () s,k dag s (Πk ) dag s (Ω(k) ) = 0 (21) Thus: P () s,k = max ( 0, dag s (Π(k) ) dag s (Ω (k) dag s (Π (k) )dag s (Ω (k) ) ) ) (22) Pluggng (22) nto (20), we obtan the dual functon: D(α (k),γ ) = { M {log dag s (Π(k) ) dag s (Ω (k) ) 1+ dag s (Ω dag s (Π (k) s=1 s,k such that dag s (Π (k) ) > dag s (Ω (k) ) > 0. (k) ) ) }+α(k) P mask (f k )+ γ K P max} (23) To solve the DP (14) for α (k) and γ, we note that the problem s convex, hence, any statonary pont s a globally optmal soluton. Moreover, as the objectve functon and constrants of the prmal problem (10) s contnuous w.r.t every entry of T, the dual functon D(α (k),γ ) s dfferentable w.r.t α (k) and γ [26]. Hence, a gradent algorthm can be used to obtan the optmal Lagrangan multplers α (k) and γ by searchng for a statonary pont of the augmented Lagrangan of DP. It should be noted that even f the dual 15

16 functon s not dfferentable (.e., multple subgradents may exst), a subgradent-based search algorthm wth approprate step sze can stll be used to converge to the optmal pont [26]. The augmented Lagrangan of DP s gven by: L(α (k),γ,p,λ (k) ) = D(α (k),γ )+ p 2 {(max{0,λ(1) pγ }) 2 (λ (1) ) 2 }+ p 2 {(max{0,λ (k+1) pα (k) }) 2 (λ (k+1) ) 2 } (24) where p s a postve penalty parameter (for volatng the constrants) and λ (k) s are nonnegatve Lagrangan multplers. Our gradent algorthm uses Armjo step wth steepest descent drecton. Ths search mechansm together wth the above analyss are summarzed n Algorthm 1. We emphasze that by explotng the strong dualty, ths algorthm needs only to deal wth K +1 varables, nstead of 2KM 2 varables of the prmal problem (10). Although the ndvdual optmzaton (10) s to be solved dstrbutedly at each node, at the acheved NE, network throughput s analytcally guaranteed to be as good as that of a locally optmal pont of the network optmzaton problem (6). Before developng a centralzed algorthm that serves as a smulaton s performance benchmark, let s brefly dscuss how a MAC protocol can mplement the Algorthm 1 and ts convergence behavor. D. A MAC Protocol Usng ether a dedcated control channel or some frequency hoppng mechansm to establsh an ntal dalogue, a MAC protocol that executes the dstrbuted Algorthm 1 can be desgned. Ths protocol dvdes the tme axs nto three wndows: Access wndow, tranng wndow, and data wndow. The access wndow s dedcated to CR nodes that have data to send. These nodes frst exchange some ntal rendezvous packets (e.g., RTS and CTS). Unlke IEEE , our MAC desgn does not use RTS/CTS packets to slence nearby nodes to reserve the transmsson floor for the comng transmsson. Instead, we use these sgnallng 16

17 Algorthm 1 Dstrbuted Algorthm for the Power Allocaton and Spectrum Management Game 1: Input: T = [ T 1 (t+1),..., T 1 (t+1), T +1 (t),..., T N (t)] wth Gauss-Sedel teraton T = [ T 1 (t),..., T 1 (t), T +1 (t),..., T N (t)] wth Jacob teraton 2: Intalze T (k) (t+1) T (k) (t),γ 0;α (k) 0, k Ψ K 3: whle true do 4: β.7,σ.1%used n Armjo search 5: λ (k).1 k = 1...(K +1) 6: p 1 7: whle L(α (k),γ,p,λ (k) ) 0 do 8: step 0.1 9: D L(α (k),γ,p,λ (k) ) 10: d step D;m 0 11: {Fnd Armjo step sze} 12: whle L(α (k),γ,p,λ (k) ) L(α (k) +d,γ +d,p,λ (k) ) σβ m step LD do 13: step step β;m m+1 14: d step D 15: end whle 16: α (k) α (k) +d,γ γ +d 17: end whle 18: 19: f mn(α (k),γ, k Ψ K ) 0 break k Ψ K : 20: λ (k) λ (k) pα (k) f λ (k) pα (k) > 0 else λ (k) = 0 21: 22: λ (1) λ (1) pγ f λ (1) pγ 0 else λ (1) = 0 p p µ%µ 1, ncrease cost of volaton 23: end whle 24: Plug γ, α (k) nto (16) (Theorem 3) to fnd T (k) P () k s found from (18), (19), (22). T (k) 25: RETURN T (k) (t+1), k Ψ K at tme (t+1) s found from (17). packets handshake a number of node pars. After ths phase, several pars of CR users communcate durng the tranng wndow, whose purpose s to exchange/negotate transmt strateges (precodng matrces). The sgnallng packets n ether the access or transmt wndows can also be used to embed tranng sequences to obtan channel gan matrces. The data wndow then follows wth multple DATA packets sent usng negotated transmsson strateges. The mechansm s referred to as a flow-based one n [2]. To reduce the feedback overhead n the tranng wndow, one may relax the tme scale of recalculatng the prcng-factor matrx. Ths represents a tradeoff between throughput and feedback freshness. One can even omt the tranng wndow by havng nodes to embed updatng nformaton nto every DATA and ACK packet. Then, upon recevng an ACK for each DATA packet sent, a transmtter recomputes ts prcng-factor matrx. Ths method s referred to as a packet-based one where less overhead and lower delay are sustaned at the expense of network throughput and convergence speed [29] [2]. An mportant ssue for protocol desgners s how to set the sze of the tranng wndow. That depends 17

18 on the convergence speed of the updatng process. To ensure that the tranng wndow s not too long, the updatng and negotaton processes 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. Though we cannot prove the convergence under the Jacob teraton, smulatons show that the dstrbuted algorthm converges faster wth Jacob teratons than Gauss-Sedel (less than nne teratons for about ten lnks n Fgure 5). The convergence behavor under the Gauss-Sedel teraton s clamed n the followng theorem. Theorem 4: Under the sequental updatng procedure (Gauss-Sedel), the dstrbuted Algorthm 1 drves the game (10) to ts NE. Proof: See Appendx IV. In [20] [18], the authors converted the throughput maxmzaton game (of a sngle-frequency MIMO CRN) nto a varatonal nequalty problem (see [30] and theren references for a tutoral on varatonal nequalty theory) for the purpose of provdng the convergence and unqueness condtons of a 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 requres transmtters to adjust ts transmsson parameters and the latter s uncertan and cannot be 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 condton. Regardng the unqueness of the NE, our game may have more than one NEs, however the performance at any acheved NE s always bounded from below by that of a locally optmal soluton of the network optmzaton problem. Both Gauss-Sedel and Jacab updatng procedures are synchronous ones that requre CR nodes have to be n synch. Ths may not be always feasble as CRs are spatally dstrbuted. By contrast, asynchronous update procedures refer to cases where some players at some teratons do not update others ther strateges (e.g., due to collsons). Usng varatonal nequalty theory, the mean-value theorem, and follow the routne n [18], we provde condtons under whch the game (10) converge to a unque NE under ether synchronous or asynchronous updates (we omt detaled manpulaton due to space lmt). 18

19 Theorem 5: The game (10) always converges to a unque NE f all channel matrces are full column-rank and the spectrum radus of matrx J 1 Γ less than 1: radus(j 1 Γ) < 1 where J = def σ σ σ N Γ = def 0 κ 1,2... κ 1,N κ 2, κ 2,N κ N,1 κ N, and def σ = mn eg 2 mn H {k}h d(), ( I+ j Φ N P mask (f k )H {k} d(j), H{k}H d(j), ) ( ( ( )) def κ j, = max eg max H {k} k Ψ d(j), H{k}H d(j), )eg max H {k}h d(), H{k} d(), K ) 1 H {k} d(), j Ψ N wth eg mn (.) and eg max (.) operators yeld the smallest and largest egenvalues of a matrx (.), respectvely. V. CENTRALIZED ALGORITHM From a game theoretc perspectve, a centralzed algorthm can be obtaned by formulatng the problem as a cooperatve game, where a network operator somehow controls the behavors of all players n order to maxmze the network throughput (total payoff). In ths secton, we use the augmented Lagrangan multpler method to derve such a centralzed algorthm. We rewrte network-wde problem (6) as follows: mnmze R () { T (k),, Φ N } Φ N s.t. c = tr( T (k) ) P max 0 Φ N c k, = tr( T (k) ) P mask (f k ) 0 k Ψ K, Φ N (25) 19

20 The augmented Lagrangan of (25) s gven by [26]: L( T,α (k),γ,p) = R () + p {(max{0,γ +pc }) 2 (γ ) 2 }+ p 2 2 Φ N Φ N Φ N {(max{0,α (k) +pc k, }) 2 (α (k) ) 2 } (26) where p s a postve penalty parameter (for volatng the constrants), and α (k) and γ are nonnegatve Lagrangan multplers. At a locally optmal soluton, we have: 0 = (k) L( T,α,γ,p) = T (k) j Φ N \{} R (k) j T (k) R(k) T (k) + p 2 { {(max{0,γ +pc }) 2 } T (k) + {(max{0,α(k) +pc k, }) 2 } } T (k) (27) The frst term n (27) s computed n (35). Its second term s gven as: R (k) T (k) = H (k)h d(), (C(k) d() +H(k) d(), T (k) H (k)h d(), ) 1 H (k) d(), T (k). (28) Snce c and c k, are contnuously dfferentable w.r.t every entry of T, the thrd and fourth terms n (27) are also contnuously dfferentable [26]. Ther dervatves are as follows: {(max{0,γ +pc }) 2 } T (k) {(max{0,α (k) +pc k, }) 2 } } = T (k) 0 f γ +pc 0 = 2p(γ +pc ) T (k) 0 f α (k) +pc k, 0 2p(α (k) +pc k, ) T (k) As mentoned earler, because the network optmzaton problem s not convex, the centralzed algorthm can only lead the network to operate at a locally optmal pont. For that purpose, we use the gradent search algorthm wth Armjo step sze [26] to fnd ( T,α (k),γ,p) such that equaton (27) holds for all frequency bands k and all users. The detals of the centralzed algorthm s presented n Algorthm 2. 20

21 We emphasze that network throughput may vary from a locally optmal pont to another. To account for such phenomenon, one can run the smulatons multple tmes wth varous startng ponts (ntalzatons) and take the average of the acheved throughput. The runnng tme of 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 wthx = [R[vec( T )] T,I[vec( T )] T] T [ = R[vec( T (1) )] T,...,R[vec( T (K) )] T,I[vec( T (1) )] T,...,I[vec( T (K) )] T] T The gradent of the correspondng Lagrangan s gven by: [ x L = 2 R[vec( L )] T L,...,R[vec( )] T,I[vec( L )] T L,...,I[vec( T (1) 1 T (K) N T (1) 1 Algorthm 2 Centralzed Algorthm for the Socal Optmzaton Problem (6) 1: Intalze T (k) I,γ 0;α (k) 0, k Ψ K, Φ N 2: whle true do 3: β.7,σ.1%used n Armjo search 4: γ 0,α (k) 0, k Ψ K, Φ N 5: p 1 6: whle L( T,α (k),γ,p) 0 do 7: step 0.1 8: D L( T,α (k),γ,p) 9: d step D;m 0 10: {Fnd Armjo step sze} 11: whle L( T,α (k),γ,p) L( T+d,α (k),γ,p) σβ m step LD do 12: step step β;m m+1 13: d step D 14: end whle 15: T T+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 (k) = α (k) +pc k, f α (k) +pc k, 0 else α (k) = 0 21: p p µ%µ 1, ncrease cost of volaton 22: end whle 23: RETURN T (k), k Ψ K, Φ N T (K) N )] T ] T VI. NUMERICAL RESULTS In ths secton, we numercally evaluate the performance of the dstrbuted algorthm usng MATLABbased smulatons. We compare the network throughput of the dstrbuted algorthm wth the centralzed one and wth a greedy algorthm, n whch nodes selfshly attempt to maxmze ther own rates. The greedy 21

22 algorthm s exactly the same as the dstrbuted one except that ts 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 [14] for each channel. We emphasze that ths unform algorthm does not meet the optmalty condtons (33) of the network problem (6) (a) Greedy algorthm: Channel (b) Greedy algorthm: Channel (c) Greedy algorthm: Channel (d) Dstrbuted algorthm: Channel 1 (e) Dstrbuted algorthm: Channel 2 (f) Dstrbuted algorthm: Channel (g) Centralzed algorthm: Channel 1 (h) Centralzed algorthm: Channel 2 () Centralzed algorthm: Channel 3 Fg. 3. Antenna radaton patterns under the greedy, dstrbuted, and centralzed algorthms. Because the number of varables n the centralzed algorthm s qute hgh (2NKM 2 ), ts runnng tme can be very long. Therefore, to compare the performances of the four algorthms, we consder a CRN of N = 10 lnks wth K = 3 bands (f 1 = 2.4 GHz, f 2 = 2.7 GHz, and f 3 = 3 GHz wth dentcal channel 22

23 bandwdth of 1 MHz), and antenna array sze M = 4. The results are averaged over 30 runs. In each run, N lnks are randomly placed n a 100 meter 100 meter square. The maxmum power at each node s 2W and the power mask s 0.8W on all frequency bands. The channel fadng s flat wth free-space attenuaton factor of 2. The spreadng angles of the sgnal at the receve antennas range from π/5 to π/5. For the lowest frequency, we assume that the receved power at a reference dstance of 100 meters reduces 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. As mentoned before, the nose from PR transmssons s treated as floor nose that together wth the thermal nose are normalzed to a unt varance. The ntalzatons of the precodng matrces are dfferent for all algorthms. A snapshot of the network topology and antenna radaton patterns (at the converged ponts) over dfferent frequences s shown n Fgure 3. We can vsually note that the transmtters under the dstrbuted and centralzed algorthms often steer ther beams away from neghborng recevers. Ths results from attemptng to mnmze the prce functon (11). It can also be seen that the antenna patterns of the dstrbuted and centralzed algorthms are very smlar, suggestng the two algorthms may converge to the same pont. Fgure 4 depcts the network throughput under four algorthms (dstrbuted, centralzed, greedy, and unform) versus the number of teratons. Though the network performance at the converged ponts for 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 almost doubles the network throughput compared wth the greedy algorthm. The unform algorthm also mproves network throughput over the greedy one but t remans nferor to the dstrbuted algorthm. Ths s because the unform algorthm evenly allocates ts power over all avalable channels and does not optmze over the frequency dmenson, whle the dstrbuted algorthm attempts to optmze the antenna radaton patterns and the power allocaton over both space and frequency. To evaluate the energy effcency of the four algorthms, we record the average power consumpton and 23

24 100 Network Throughput (Mbps) Iteratons Dstrbuted Greedy Centralzed Unform Fg. 4. Network throughput vs. teratons. power allocaton over all nodes and all run n Table I. As shown n Table I, wthout regulatng nterference, nodes under the greedy algorthm selfshly compete for ther own throughput by always usng ther maxmum power (2W ), leadng to the hghest power consumpton among the four algorthms. The power consumpton for the dstrbuted algorthm s comparable to that of the centralzed and unform algorthms, 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 n Table II. From Tables I and II, we note 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. Channels Centralzed Greedy Dstrbuted Unform f f f Total (W) TABLE I AVERAGE POWER CONSUMPTION AND POWER ALLOCATION OVER DIFFERENT CHANNELS (IN WATTS). Antennas f 1 f 2 f e e Total=1.913(W) TABLE II POWER ALLOCATION AT A NODE OVER SPACE AND FREQUENCY DIMENSIONS UNDER THE DISTRIBUTED ALGORITHM (IN WATTS). 24

25 We say that the algorthm converges f the change n the throughput of one teraton (relatve to the prevous teraton) s less than a gven threshold (.e., 3%). The convergence speed of the dstrbuted algorthm versus the number of lnks s shown n Fgure Iteratons to Converge Number of Lnks Jacob Gauss Sedel Fg. 5. Convergence speed of the dstrbuted algorthm. Fgure 6 depcts the network throughput under the dstrbuted and greedy algorthms versus the number of lnks usng Jacob update polcy. The dstrbuted algorthm consstently mproves the throughput over the greedy one. The mprovement becomes more sgnfcant wth a hgher number of lnks. That s because as the node densty ncreases (hgher number of lnks), network nterference ncreases. Interference management becomes more crtcal and more mpact on throughput. 180 Network Throughput (Mbps) Number of Lnks Dstrbuted Greedy Fg. 6. Network throughput vs. the number of lnks. 25

26 VII. CONCLUSIONS In ths work, we nvestgated the spectrum sharng problem n mult-antenna CRNs. By adjustng the precodng matrces, we allocate power over both the frequency and space dmensons whle managng the antenna s radaton beams to reduce network nterference, amng at maxmzng the network throughput. Usng game theory and the strong dualty n convex optmzaton, we desgned a low-complexty dstrbuted algorthm and ts correspondng MAC protocol that acheve 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 showed that the proposed algorthm dramatcally mproves network throughput and acheves hgher energy effcency, compared wth exstng solutons. REFERENCES [1] 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 [2] 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 , [3] D. Tse and P. Vswanath, Fundamentals of Wreless Communcaton. Cambrdge Unversty Press, May [4] 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 [5] K. Sundaresan and R. Svakumar, Routng n ad-hoc networks wth MIMO lnks, n Proceedngs of the ICNP Conference, Nov [6] S. Chu and X. Wang, Opportunstc and cooperatve spatal multplexng n MIMO ad hoc networks, n Proceedngs of MOBIHOC Conference, 2008, pp [7] J. Lu, Y. Hou, Y. Sh, and H. Sheral, On performance optmzaton for mult-carrer MIMO ad hoc networks, n Proceedngs of the ACM MOBIHOC Conference, May 2009, pp [8] U. Phuyal, A. Punchhewa, V. Bhargava, and C. Despns, Power loadng for multcarrer cogntve rado wth MIMO antennas, n Proceedngs of the IEEE Wreless Communcatons and Networkng Conference (WCNC), Aprl 2009, pp [9] 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 , [10] C. Sh, R. Berry, and M. Hong, Local nterference prcng for dstrbuted beamformng n MIMO networks, n Proceedngs of IEEE Mltary Communcatons Conference MILCOM, 2009, pp

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