Cognitive MIMO Radio: A Competitive Optimality Design Based on Subspace Projections
Cognitive MIMO Radio: A Competitive Optimality Design Based on Subspace Projections
Authors: Gesualdo Scutari, Daniel P. Palomar, Sergio Barbarossa
Cognitiv e MIMO Radio: A Comp etitiv e Optimalit y Design Based on Subspae Pro jetions Gesualdo Sutari 1 , Daniel P . P alomar 2 , and Sergio Barbarossa 1 E-mail: { sutari, sergio } info om.uniroma1.it, palomarust.hk 1 Dpt. INF OCOM, Univ. of Rome La Sapienza, Via Eudossiana 18, 00184 Rome, Italy 2 Dpt. of Eletroni and Computer Eng., Hong K ong Univ. of Siene and T e hnology , Hong K ong. Submitted to IEEE Signal Pr o essing Magazine , Mar h 17, 2008. A epted July 24, 2008. 1 In tro dution Radio regulatory b o dies are reen tly reognizing that rigid sp etrum assignmen t gran ting exlusiv e use to liensed servies is highly ineien t., due to the high v ariabilit y of the tra statistis aross time, spae, and frequeny . Reen t F ederal Comm uniations Commission (F CC) measuremen ts sho w that, in fat, the sp etrum usage is t ypially onen trated o v er ertain p ortions of the sp etrum, while a signian t amoun t of the liensed bands (or idle slots in stati time division m ultiple aess systems with burst y tra) remains un used or underutilized for ninet y p eren t of time [1℄. It is not surprising then that this ineieny is motiv ating a urry of resear h ativities in engineering, eonomis and regulation omm unities in the eort of nding more eien t sp etrum managemen t p oliies. As p oin ted out in man y reen t w orks [2, 3, 4, 5℄, the most appropriate approa h to ta kle the great sp etrum v ariabilit y as a funtion of time and spae alls for dynami aess strategies that adapt to the eletromagneti en vironmen t. Cognitiv e Radio (CR) originated as a p ossible solution to this problem [6℄ obtained b y endo wing the radio no des with ognitiv e apabilities, e.g., the abilit y to sense the eletromagneti en vironmen t, mak e short term preditions, and reat in telligen tly in order to optimize the usage of the a v ailable resoures. Multiple paradigms asso iated with CR ha v e b een prop osed [ 2 , 3 , 4 , 5 ℄, dep ending on the p oliy to b e follo w ed with resp et to the li ense d users, i.e. the users who ha v e aquired the righ t to transmit o v er sp ei p ortions of the sp etrum buying the relativ e liense. The most ommon strategies adopt a hierar hial aess struture, distinguishing b et w een primary users, or legay sp etrum holders, and se ondary users, who aess the liensed sp etrum dynamially , under the onstrain t of not induing Qualit y of Servie (QoS) degradations in tolerable to the primary users. Within this on text, three basi approa hes ha v e b een onsidered to allo w onurren t omm uniations: sp e trum overlay, underlay and interwe ave . 1 1 There is no strit onsensus on some of the basi terminology in ognitiv e systems [ 4 ℄. Here w e use in terw ea v e as in [5℄ whi h is sometimes referred to as o v erla y omm uniations [4 ℄. 1 In o v erla y systems, as prop osed in [7℄, seondary users allo ate part of their p o w er for seondary transmissions and the remainder to assist (rela y) the primary transmissions. By exploiting sophistiated o ding te hniques su h as dirt y pap er o ding based on the kno wledge of the primary users' message and/or o deb o ok at the ognitiv e transmitter, these systems oer the p ossibilit y of onurren t transmissions without apait y p enalties. Ho w ev er, although in teresting from an information theoreti p ersp etiv e, these te hniques are diult to implemen t as they require nonausal kno wledge of the primary signals at the ognitiv e transmitters. In underla y systems, the seondary users are also allo w ed to share resoures with the primary users, but without an y kno wledge ab out the primary users' signals and under the strit onstrain t that the sp etral densit y of their transmitted signals fall b elo w the noise o or at the primary reeiv ers. This in terferene onstrain t an b e met using spread sp etrum or ultra-wideband omm uniations from the seondary users. Both transmission te hniques do not require the estimation of the eletromagneti en vironmen t from seondary users, but they are mostly appropriate for short distane omm uniations, b eause of the strong onstrain ts imp osed on the maxim um p o w er radiated b y the seondary users. Con v ersely , in terw ea v e omm uniations, initially en visioned in [ 6℄, are based on an opp ortunisti or adaptiv e usage of the sp etrum, as a funtion of its real utilization. Seondary users are allo w ed to adapt their p o w er allo ation as a funtion of time and frequeny , dep ending on what they are able to sense and learn from the en vironmen t, in a nonin trusiv e manner. Rather than imp osing a sev ere onstrain t on their transmit p o w er sp etral densit y , in in terw ea v e systems, the seondary users ha v e to gure out when and wher e to transmit. Dieren tly from underla y systems, this opp ortunisti sp etrum aess requires an opp ortunit y iden tiation phase, through sp etrum sensing, follo w ed b y an opp ortunit y exploitation mo de [4℄. F or a fasinating motiv ation and disussion of the signal pro essing hallenges faed in in terw ea v e ognitiv e radio systems, w e suggest the in terested reader to refer to [ 2℄. In this pap er w e fo us on opp ortunisti resoure allo ation te hniques in hierar hial ognitiv e net- w orks, as they seem to b e the most suitable for the urren t sp etrum managemen t p oliies and legay wireless systems [4℄. W e are sp eially in terested in devising the most appropriate form of onurren t omm uniations of ognitiv e users omp eting o v er the ph ysial resoures let a v ailable from primary users. Lo oking at opp ortunisti omm uniation paradigm from a broad signal pro essing p ersp etiv e, the se- ondary users are allo w ed to transmit o v er a m ulti-dimensional spae, whose o ordinates represen t time slots, frequeny bins and (p ossibly) angles, and their goal is to nd out the most appropriate transmission strategy , assuming a giv en p o w er budget at ea h no de, exploring all a v ailable degrees of freedom, under the onstrain t of induing a limited in terferene, or no in terferene at all, at the primary users. In general, the optimization of the transmission strategies requires the presene of a en tral no de ha ving full kno wledge of all the hannels and in terferene struture at ev ery reeiv er. But this p oses a serious implemen tation problem in terms of salabilit y and amoun t of signaling to b e ex hanged among the no des. The required extra signaling ould, in the end, jeopardize the promise for higher eieny . T o o v erome this diult y , w e onen trate on deen tralized strategies, where the ognitiv e users are able 2 to self-enfore the negotiated agreemen ts on the sp etrum usage without the in terv en tion of a en tralized authorit y . The philosoph y underlying this approa h is a omp etitive optimality riterion, as ev ery user aims for the transmission strategy that unilaterally maximizes his o wn pa y o funtion. The presene of onurren t seondary users omp eting o v er the same resoures adds dynamis to the system, as ev ery seondary user will dynamially reat to the strategies adopted b y the other users. The main question is then to establish whether, and under what onditions, the o v erall system an ev en tually on v erge to an equilibrium from whi h ev ery user is not willing to unilaterally mo v e, as this w ould determine a p erformane loss. This form of equilibrium oinides with the w ell-kno wn onept of Nash Equilibrium (NE) in game theory (see, e.g., [8 , 9℄). In fat, game theory is the natural to ol to devise deen tralized strategies allo wing the seondary users to nd out their b est resp onse to an y giv en hannel and in terferene senario and to deriv e the onditions for the existene and uniqueness of NE. Within this on text, in this pap er, w e prop ose and analyze a totally deen tralized approa h to design ognitiv e MIMO transeiv ers, satisfying a omp etitiv e optimalit y riterion, based on the a hiev emen t of Nash equilibria. T o tak e full adv an tage of all the opp ortunities oered b y wireless omm uniations, w e assume a fairly general MIMO setup, where the m ultiple hannels ma y b e frequeny hannels (as in OFDM systems) [10℄-[12 ℄, time slots (as in TDMA systems) [ 10, 11℄, and/or spatial hannels (as in transmit/reeiv e b eamforming systems) [13℄. Whenev er a v ailable, m ultiple an tennas at the seondary transmitters ould b e used, for example, to put n ulls in the an tenna radiation pattern of seondary transmitters along the diretions iden tifying the primary reeiv ers, th us enabling the share of frequeny and time resoures with no additional in terferene. Our initial goal is to pro vide onditions for the existene and uniqueness of NE p oin ts in a game where seondary users omp ete against ea h other to maximize their p erformane, under the onstrain t on the maxim um (or n ull) in terferene indued on the primary users. The next step is then to desrib e lo w-omplex totally distributed te hniques able to rea h the equilibrium p oin ts of the prop osed games, with no o ordination among the seondary users. 2 System Mo del: Cognitiv e Radio Net w orks W e onsider a senario omp osed b y heterogeneous wireless systems (primary and seondary users), as illustrated in Figure 1 . The setup ma y inlude p eer-to-p eer links, m ultiple aess, or broadast hannels. The systems o existing in the net w ork do not ha v e a ommon goal and do not o op erate with ea h other. Moreo v er, no en tralized authorit y is assumed to handle the net w ork aess from seondary users. Th us, the seondary users are allo w ed, in priniple, to omp ete for the same ph ysial resoures, e.g., time, frequeny , and spae. W e are in terested in nding the optimal transmission strategy for the seondary users, using a deen tralized approa h. A fairly general system mo del to desrib e the signals reeiv ed b y the seondary users is the Gaussian ve tor in terferene hannel: y q = H q q x q + X r 6 = q H r q x r + n q , (1) 3 Figure 1: Hierar hial ognitiv e radio net w ork with primary and seondary users. where x q is the n T q -dimensional blo k of data transmitted b y soure q , H q q is the n R q × n T q (omplex) hannel matrix b et w een the q -th transmitter and its in tended reeiv er, H r q is the n R q × n T r ross- hannel matrix b et w een soure r and destination q , y q is the n R q -dimensional v etor reeiv ed b y destination q , and n q is the n R q -dimensional noise plus in terferene v etor. The rst term in the righ t-hand side of ( 1) is the useful signal for link q , the seond and third terms represen t the Multi-User In terferene (MUI) reeiv ed b y seondary user q and aused from the other seondary users and the primary users, resp etiv ely . The v etor n q is assumed to b e zero-mean irularly symmetri omplex Gaussian with arbitrary (nonsingular) o v ariane matrix R n q . F or the sak e of simpliit y and la k of spae, w e onsider here only the ase where the hannel matries H q q are square nonsingular. W e assume that ea h reeiv er is able to estimate the hannel from its in tended transmitter and the o v erall MUI o v ariane matrix (alternativ ely , to mak e short term preditions, with negligible error). 2 The reeiv er sends then this information ba k to the transmitter through a lo w bit rate (error-free) feedba k hannel, to allo w the transmitter to ompute the optimal transmission strategy o v er its o wn link. The mo del in (1) represen ts a fairly general MIMO setup, desribing m ultiuser transmissions o v er m ultiple hannels, whi h ma y represen t frequeny hannels (as in OFDM systems) [ 10℄-[12 ℄, time slots (as in TDMA systems) [10 , 11 ℄, or spatial hannels (as in transmit/reeiv e b eamforming systems) [13℄. Dif- feren tly from traditional stati or en tralized sp etrum assignmen t, the ognitiv e radio paradigm enables seondary users to transmit with o v erlapping sp etrum and/or o v erage with primary users, pro vided that the degradation indued on the primary users' p erformane is n ull or tolerable. Ho w to imp ose in ter- ferene onstrain ts on seondary users is a omplex and op en regulatory issue [2, 4℄. Roughly sp eaking, restritiv e onstrain ts ma y marginalize the p oten tial gains oered b y the dynami resoure assignmen t me hanism, whereas lo ose onstrain ts ma y aet the ompatibilit y with legay systems. Both determinis- ti and probabilisti in terferene onstrain ts ha v e b een suggested in the literature [1, 2, 4 , 15 ℄, namely: the 2 Ho w to obtain b oth hannel-state information and MUI o v ariane matrix estimation go es b ey ond the sop e of this pap er; the in terested reader ma y refer to, e.g., [ 2, 4℄, where lassial signal pro essing estimation te hniques are prop erly mo died to b e suessfully applied in a ognitiv e radio en vironmen t. 4 maxim um MUI in terferene p o w er lev el p ereiv ed b y an y ativ e primary user (the so-alled interfer en e temp er atur e limit ) [1, 2℄ and the maxim um probabilit y that the MUI in terferene lev el at ea h primary user's reeiv er ma y exeed a presrib ed threshold [4, 15℄. In the presene of sensing errors, the aess to hannels iden tied as idle should also dep end on the go o dness of the hannel estimation. As sho wn in [17℄, in this ase the optimal strategy is probabilisti, with an probabilit y dep ending on b oth the false alarm and miss probabilities. In this pap er w e are primarily in terested in analyzing the on ten tion among the seondary users o v er a m ultiuser hannel where there are primary users as w ell. T o limit the omplexit y of the problem, in the eort to nd out distributed te hniques guaran teed to on v erge to NE p oin ts, w e restrit our analysis to onsider only deterministi in terferene onstrain ts, alb eit expressed in a v ery general form. In partiular, w e en visage the use of the follo wing p ossible in terferene onstrain ts (see also Figure 2 ): Co.1 Maximum tr ansmit p ower for e ah tr ansmitter : E n k x q k 2 2 o = T r ( Q q ) ≤ P q , (2) where Q q denotes the o v ariane matrix of the sym b ols transmitted b y user q and P q is the transmit p o w er in units of energy p er transmission. Co.2 Nul l onstr aints : U H q Q q = 0 , (3) where U q is a strit tall matrix (to a v oid the trivial solution Q q = 0 ), whose olumns represen t the spatial and/or the frequeny diretions along with user q is not allo w ed to transmit. W e assume, without loss of generalit y (w.l.o.g.), that ea h matrix U q is full-olumn rank. Co.3 Soft shaping onstr aints : T r G H q Q q G q ≤ P a ve q , (4) where the matries G q are su h that their range spae iden ties the subspae where the in terferene lev el should b e k ept under the required threshold. 3 Co.4 Pe ak p ower onstr aints : the a v erage p eak p o w er of ea h user q an b e on trolled b y onstraining the maxim um eigen v alue [denoted b y λ max ( · ) ℄ of the transmit o v ariane matrix along the diretions spanned b y the olumn spae of G q : λ max G H q Q q G q ≤ P peak q , (5) where P peak q is the maxim um p eak p o w er that an b e transmitted along the spatial and/or the frequeny diretions spanned b y the olumn spae of G q . 3 The in terferene temp erature limit onstrain t [2 ℄ is giv en b y the aggregated in terferene indued b y all seondary users. In this pap er, w e assume that the primary user imp osing the soft onstrain t, has already omputed the maxim um tolerable in terferene p o w er P a ve q for ea h seondary user. The p o w er limit P a ve q an also b e the result of a negotiation or opp ortunisti based pro edure b et w een primary users (or regulatory agenies) and seondary users. 5 Figure 2: Example of n ull/soft shaping onstrain ts. The struture of the n ull onstrain ts in (3) is a v ery general form to express the strit limitation imp osed on seondary users to prev en t them from transmitting o v er the sub hannels o upied b y the primary users. These sub hannels are mo deled as v etors b elonging to the subspae spanned b y the olumns of ea h matrix U q . This form inludes, as partiular ases, the imp osition of n ulls o v er: 1) the frequeny bands o upied b y the primary reeiv ers; 2) the time slots o upied b y the primary users; 3) the angular diretions iden tifying the primary reeiv ers as observ ed from the seondary transmitters. In the rst ase, the subspae is spanned b y a set of IFFT v etors, in the seond ase b y a set of anonial v etors, and in the third ase b y the set of steering v etors represen ting the diretions of the primary reeiv ers as observ ed from the seondary transmitters. It is w orth emphasizing that the struture of the n ull onstrain ts in (3 ) is m u h more general than the three ases men tioned ab o v e, as it an inorp orate an y om bination of the frequeny , time and spae o ordinates. The use of the spatial domain an greatly impro v e the apabilities of ognitiv e users, as it allo ws them to transmit o v er the same frequeny band but without in terfering. This is p ossible if the seondary transmitters ha v e an an tenna arra y and use a b eamforming that puts n ulls o v er the diretions iden tifying the primary reeiv ers. Of ourse, this requires the iden tiation of the primary reeiv ers, a task that is m u h more demanding than the detetion of primary transmitters [4℄. As an example, there are some reen t w orks sho wing that, in the appliation of CR o v er the sp etrum allo ated to ommerial TV, one migh t exploit the lo al osillator leak age p o w er emitted b y the RF fron t end of the TV reeiv er to lo ate the reeiv ers [18 ℄. Of ourse, in su h a ase, the detetion range is quite short and this alls for a deplo ymen t of sensors v ery lose to the p oten tial reeiv ers. A dieren t senario p ertains to ellular systems. In su h a ase, the mobile users migh t b e rather hard to lo ate and tra k. Ho w ev er, the base stations are relativ ely easier to iden tify . Hene, in a ellular system op erating in a time-division duplexing (TDD) mo de, the seondary users ould exploit the time slot allo ated for the uplink hannel and put a n ull in the diretion of the base stations. This w ould a v oid an y in terferene to w ards the ellular system 6 users, without the need of tra king the mobile users. The soft shaping onstrain ts expressed in (4 ) and (5) represen t a onstrain t on the total a v erage and p eak a v erage p o w er radiated (pro jeted) along the diretions spanned b y the olumn spae of matrix G q . They are a relaxed form of (3) and an b e used to k eep the p ortion of the in terferene temp erature generated b y ea h seondary user q under the desired v alue. In fat, under (4 )-(5), the seondary users are allo w ed to transmit o v er some sub hannels o upied b y the primary users, but only pro vided that the in terferene that they generate falls b elo w a presrib ed threshold. F or example, in a MIMO setup, the matrix G q in (4) w ould on tain, in its olumns, the steering v etors iden tifying the diretions of the primary reeiv ers. Within the assumptions made ab o v e, in v oking the apait y expression for the single user Gaussian MIMO hannel − a hiev able using random Gaussian o des b y all the users − the maxim um information rate on link q for a giv en set of users' o v ariane matries Q 1 , . . . , Q Q , is [19℄ R q ( Q q , Q − q ) = log det I + H H q q R − 1 − q H q q Q q (6) where R − q , R n q + X r 6 = q H r q Q r H H r q (7) is the MUI plus noise o v ariane matrix observ ed b y user q and Q − q , ( Q r ) r 6 = q is the set of all the users' o v ariane matries, exept the q -th one. Observ e that R − q dep ends on the strategies Q − q of the other pla y ers. 3 Resoure Sharing among Seondary Users based on Game Theory Giv en the m ultiuser nature of the senario desrib ed ab o v e, the design of the optimal transmission strate- gies of seondary users w ould require a m ultiob jetiv e form ulation of the optimization problem, as the information rate a hiev ed on ea h seondary user's link onstitutes a dieren t single ob jetiv e fun- tion. The globally optimal solutions of su h a problem − the P areto optimal surfae of the m ultiob jetiv e problem − w ould dene the largest rate region a hiev able b y seondary users, giv en the p o w er onstrain ts Co.1 - Co.4 : the rate v etor prole R ( Q ⋆ ) , [ R 1 ( Q ⋆ ) , . . . , R Q ( Q ⋆ )] is P areto optimal if there exists no other rate prole R ( Q ) that dominates R ( Q ⋆ ) omp onen t-wise, i.e., R ( Q ⋆ ) ≥ R ( Q ) , for all feasible Q 's, where at least one inequalit y is strit. Unfortunately , the omputation of the rate region is analytially in tratable and th us not appliable in a ognitiv e radio senario, sine ev ery salar/m ultiob jetiv e optimization problem in v olving the rates of seondary users in (6) is not on v ex (implied from the fat that the rates R q ( Q ) are nonona v e funtions of the o v ariane matries Q ). F urthermore, ev en in the simpler ase of transmissions o v er SISO parallel hannels, the net w ork utilit y maximization (NUM) problem based on the rates funtions (6) has b een pro v ed in [24℄ to b e a strongly NP-hard problem, under v arious pratial settings as w ell as dieren t 7 hoies of the system utilit y funtion (e.g., sum-rate, w eigh ted sum-rate, geometri rate-mean). Roughly sp eaking, this means that there is no hop e to obtain an algorithm, ev en en tralized, that an eien tly ompute the exat globally optimal solution. Although in theory , the rate region ould b e still found b y an exhaustiv e sear h through all p ossible feasible o v ariane matries, the omputational omplexit y of this approa h is prohibitiv ely high, giv en the large n um b er of v ariables and users in v olv ed in the optimization. The situation is partiularly ritial in CR systems, where the ognitiv e users sense a v ery large sp etrum. Consequen tly , sub optimal algorithms ha v e b een prop osed in the literature to solv e sp eial ases of the prop osed optimization [20 ℄-[23 ℄, most of them dealing with the maximization of the (w eigh ted) sum-rate in SISO frequeny-seletiv e in terferene hannels (obtained from our general mo del when the hannel matries are diagonal, the o v ariane matries redue to the p o w er allo ation v etors, and the n ull/soft shaping onstrain ts are remo v ed) [20 , 21℄. Due to the nonon v ex nature of the problem, these algorithms either la k global on v ergene or ma y on v erge to p o or sp etrum sharing strategies. F urthermore, ev en if one deides to emplo y a sub optimal metho d, e.g., [ 20 ℄-[23℄, the algorithms are not suitable for CR systems as they are en tralized and th us annot b e implemen ted in a distributed w a y . These te hniques require a en tral authorit y (or no de in the net w ork) with kno wledge of the (diret and ross-) hannels to ompute all the transmission strategies for the dieren t no des and then to broadast the solution. This s heme w ould learly p ose a serious implemen tation problem in terms of salabilit y of the net w ork and amoun t of signaling to b e ex hanged among the no des, whi h mak es su h an approa h not app ealing in the senario onsidered in this pap er. T o o v erome the ab o v e diulties and rea h a b etter trade-o b et w een p erformane and omplexit y , w e shift our fo us to a dieren t notion of optimalit y: the omp etitiv e optimalit y riterion; whi h motiv ates a game theoretial form ulation of the system design. Using the onept of NE as the omp etitiv e optimalit y riterion, the resoure allo ation problem among seondary users is then ast as a strategi nono op erativ e game, in whi h the pla y ers are the seondary users and the pa y o funtions are the information rates on ea h link: Ea h seondary user q omp etes against the others b y ho osing the transmit o v ariane matrix Q q (i.e., his strategy) that maximizes his o wn information rate R q ( Q q , Q − q ) in (6 ), giv en onstrain ts imp osed b y the presene of the primary users, b esides the usual onstrain t on transmit p o w er. A NE of the game is rea hed when ea h user, giv en the strategy proles of the others, do es not get an y rate inrease b y unilaterally hanging his o wn strategy . The rst question to answ er under su h framew ork is whether su h an o v erall dynamial system an ev en tually on v erge to an equilibrium p oin t, while preserving the QoS of primary users. The seond basi issue is if the optimal strategies to b e adopted b y ea h user an b e omputed in a totally deen tralized w a y . W e address b oth questions in the forthoming setions. F or the sak e of simpliit y , w e start onsidering only onstrain ts Co.1 and Co.2 . These onstrain ts are suitable to mo del in terw ea v e omm uniations among seondary users where, in general, there are restritions on when and where they ma y transmit (this an b e done using the n ull onstrain ts Co.2 ). Then, w e allo w underla y and in terw ea v e omm uniations sim ultaneously , b y inluding in the optimization also in terferene onstrain ts Co.3 and Co.4 . 8 3.1 Rate maximization game with n ull onstrain ts Giv en the rate funtions in (6 ) and onstrain ts Co.1 - Co.2 , the rate maximization game is formally dened as: ( G 1 ) : maximize Q q 0 R q ( Q q , Q − q ) sub ject to T r ( Q q ) ≤ P q , U H q Q q = 0 ∀ q = 1 , · · · , Q, (8) where Q is the n um b er of pla y ers (the seondary users) and R q ( Q q , Q − q ) is the pa y o funtion of pla y er q , dened in (6). Without the n ull onstrain ts, the solution of ea h optimization problem in (8 ) w ould lead to the w ell-kno wn MIMO w aterlling solution [19 ℄. The presene of the n ull onstrain ts mo dies the problem and the solution for ea h user is not neessarily a w aterlling an ymore. Nev ertheless, w e sho w no w that in tro duing a prop er pro jetion matrix the solutions of (8 ) an still b e eien tly omputed via a w aterlling-lik e expression. T o this end, w e rewrite game G 1 in a more on v enien t form as detailed next. In tro duing the pro jetion matrix P R ( U q ) ⊥ = I − U q ( U H q U q ) − 1 U H q (the orthogonal pro jetion on to R ( U q ) ⊥ , where R ( · ) is the range spae op erator), it follo ws from the onstrain t U H q Q q = 0 that an y optimal Q q in (8 ) will alw a ys satisfy: Q q = P R ( U q ) ⊥ Q q P R ( U q ) ⊥ . (9) The game G 1 an then b e equiv alen tly rewritten as: maximize Q q 0 log det I + ˜ H H q q ˜ R − 1 − q ˜ H q q Q q sub ject to T r ( Q q ) ≤ P q Q q = P R ( U q ) ⊥ Q q P R ( U q ) ⊥ ∀ q = 1 , · · · , Q, (10) where ea h ˜ H r q , H r q P R ( U r ) ⊥ is a mo died hannel and ˜ R − q , R n q + P r 6 = q ˜ H r q Q r ˜ H H r q . A t this p oin t, the problem an b e further simplied b y noting that the onstrain t Q q = P R ( U ⊥ q ) Q q P R ( U ⊥ q ) in (10 ) is redundan t. The nal form ulation then b eomes: maximize Q q 0 log det I + ˜ H H q q ˜ R − 1 − q ˜ H q q Q q sub ject to T r ( Q q ) ≤ P q ∀ q = 1 , · · · , Q. (11) This is due to the fat that, for an y user q , an y optimal solution Q ⋆ q in (11) − the MIMO w aterlling solution [13℄ − will b e orthogonal to the n ull spae of ˜ H q q , whatev er ˜ R − q is, implying Q ⋆ q = P R ( U q ) ⊥ Q ⋆ q P R ( U q ) ⊥ . Building on the equiv alene of (8 ) and (11 ), w e an apply the results in [13 ℄ to the game in ( 11) and deriv e the struture of the Nash equilibria of game G 1 , as detailed next. Nash equilibria of game G 1 : Game G 1 alw a ys admits a NE, for an y set of hannel matries, transmit p o w er of the users, and n ull onstrain ts, sine it is a ona v e game (the pa y o of ea h pla y er is a ona v e funtion in his o wn strategy and ea h admissible strategy set is on v ex and ompat) [13 ℄. Moreo v er, it follo ws from (11) that all the Nash equilibria of G 1 satisfy the follo wing set of nonlinear matrix-v alue xed-p oin t equations [13℄: Q ⋆ q = ˜ WF q ˜ H H q q R − 1 − q ( Q ⋆ − q ) ˜ H q q , W ⋆ q Diag p ⋆ q W ⋆H q , ∀ q = 1 , · · · , Q, (12) 9 where w e made expliit the dep endene of R − q on Q ⋆ − q as R − q ( Q ⋆ − q ) ; the W ⋆ q = W q ( Q ⋆ − q ) is the semi-unitary matrix with olumns equal to the eigen v etors of matrix ˜ H H q q R − 1 − q ( Q ⋆ − q ) ˜ H q q orresp onding to the p ositiv e eigen v alues λ ⋆ q ,k = λ q ,k ( Q ⋆ − q ) , with R − q ( Q − q ) dened in (7); and the p o w er allo ation p ⋆ q = p q ( Q ⋆ − q ) satises the follo wing sim ultaneous w aterlling equation: for all k and q , p ⋆ q ( k ) = µ q − 1 λ ⋆ q ,k ! + , (13) with ( x ) + , max(0 , x ) and µ q hosen to satisfy the p o w er onstrain t P k p ⋆ q ( k ) = P q . In terestingly , the solution (12) sho ws that the n ull onstrain ts in the transmissions of seondary users an b e handled without aeting the omputational omplexit y: The optimal transmission strategy of ea h user q an b e eien tly omputed via a MIMO w aterlling solution, pro vided that the original hannel matrix H q q is replaed b y ˜ H q q . This result has an in tuitiv e in terpretation: T o guaran tee that ea h user q do es not transmit o v er a giv en subspae (spanned b y the olumns of U q ), whihever the strategies of the other users are, while maximizing his information rate, one only needs to indue in the hannel matrix H q q a n ull spae that oinides with the subspae where the transmission is not allo w ed. This is preisely what is done b y in tro duing the mo died hannel ˜ H q q . The w aterlling-lik e struture of the Nash equilibria as giv en in (12 ) along with the in terpretation of the MIMO w atelling solution as a matrix pro jetion on to a prop er on v ex set as giv en in [13 ℄ pla y a k ey role in studying the uniqueness of the NE and in deriving onditions for the on v ergene of the distributed algorithms desrib ed in Setion 4. The analysis of the uniqueness of the NE go es b ey ond the sop e of this pap er and it is addressed in [14 ℄. What is imp ortan t to remark here is that, as exp eted, the onditions guaran teeing the uniqueness of the NE imp ose a onstrain t on the maxim um lev el of MUI generated b y seondary users that ma y b e tolerated in the net w ork. But, in terestingly , the uniqueness of the equilibrium is not aeted b y the in terferene generated b y the primary users. 3.2 Rate maximization game with n ull onstrain ts via virtual noise shaping In this setion, w e sho w that an alternativ e approa h to imp ose n ull onstrain ts Co.2 on the transmissions of seondary users passes through the in tro dution of virtual in terferers. The idea b ehind this alterna- tiv e approa h an b e easily understo o d if one onsiders the transmission o v er SISO frequeny-seletiv e hannels, where all the hannel matries ha v e the same eigen v etors (the FFT v etors): to a v oid the use of a giv en sub hannel, it is suien t to in tro due a virtual noise with suien tly high p o w er o v er that sub hannel. The same idea annot b e diretly applied to the MIMO ase, as arbitrary MIMO hannel matries ha v e dieren t righ t/left singular v etors from ea h other. Nev ertheless, w e sho w ho w to design the o v ariane matrix of the virtual noise (to b e added to the noise o v ariane matrix of ea h seondary reeiv er), so that the all the Nash equilibria of the game satisfy the n ull onstrain t Co.2 along the sp eied diretions. 10 Let us onsider the follo wing strategi nono op erativ e game: ( G α ) : maximize Q q 0 log det I + H H q q R − 1 − q ,α H q q Q q sub ject to T r ( Q q ) ≤ P q ∀ q = 1 , · · · , Q, (14) where R − q ,α , R − q + α ˆ U q ˆ U H q = R n q + X r 6 = q H r q Q r H H r q + α ˆ U q ˆ U H q , (15) denotes the MUI-plus-noise o v ariane matrix observ ed b y seondary user q , plus the o v ariane matrix α ˆ U q ˆ U H q of the virtual in terferene along R ( ˆ U q ) , where ˆ U q is a tall matrix and α is a p ositiv e onstan t. Our in terest is on deriving the asymptoti prop erties of the solutions of G α , as α → + ∞ . T o this end, w e in tro due the follo wing in termediate denitions rst. F or ea h q , dene the tall matrix ˆ U ⊥ q su h that R ( ˆ U ⊥ q ) = R ( ˆ U q ) ⊥ , and the mo died hannel matries ˆ H r q = ˆ U ⊥ H q H r q ∀ r , q = 1 , · · · , Q. (16) W e then in tro due the auxiliary game G ∞ , dened as: ( G ∞ ) : maximize Q q 0 log det I + ˆ H H q q ˆ R − 1 − q ˆ H q q Q q sub ject to T r ( Q q ) ≤ P q ∀ q = 1 , · · · , Q, (17) where ˆ R − q , ˆ U ⊥ H q R n q ˆ U ⊥ q + X r 6 = q ˆ H r q Q r ˆ H H r q . (18) It an b e sho wn that games G α and G ∞ are asymptotially equiv alen t in the sense sp eied next. Nash equilibria of games G α and G ∞ : Games G α and G ∞ alw a ys admit a NE, for an y set of hannel matries, p o w er onstrain ts, and α > 0 . Moreo v er, under mild onditions guaran teeing the uniqueness of the NE of b oth games (denoted b y Q ⋆ α and Q ⋆ ∞ , resp etiv ely), w e ha v e: lim α →∞ Q ⋆ α = Q ⋆ ∞ , (19) i.e., the NE of G α asymptotially oinides with that of G ∞ . Observ e that, similarly to game G 1 , also in games G α and G ∞ , the b est-resp onse of ea h pla y er an b e eien tly omputed via MIMO w aterlling-lik e solutions, and the Nash equilibria of b oth games satisfy a sim ultaneous w aterlling equation. Using (19), one an deriv e the asymptoti prop erties of the (unique) NE of game G α as α → ∞ , through the prop erties of the equilibrium Q ⋆ ∞ of G ∞ . F ollo wing a similar approa h as in Setion 3.1 , one an sho w that ea h Q ⋆ q , ∞ satises the follo wing ondition U H q Q ⋆ q , ∞ = 0 , with U q , H − 1 q q ˆ U q . (20) Condition (20 ) pro vides, for ea h user q , the desired relationship b et w een the diretions of the virtual noise to b e in tro dued in the noise o v ariane matrix of the user (see (18 )) − the matrix ˆ U q − and the real 11 diretions along with user q will not allo ate an y p o w er, i.e., the matrix U q . It turns out that if user q is not allo w ed to allo ate p o w er along U q , it is suien t to ho ose in (18) ˆ U q , H q q U q . Sine the existene and uniqueness of the NE of game G α do not dep end on α , the (unique) NE of G α (that in general will dep end on the v alue of α ) an b e rea hed using the asyn hronous algorithms desrib ed in Setion 4, irresp etiv e of the v alue of α . Th us, for suien tly large v alues of α , the NE of G α tends to satisfy ondition (20 ); whi h pro vides an alternativ e w a y to imp ose onstrain t Co.2 . 3.3 Rate maximization game with soft and n ull onstrain ts W e fo us no w on the rate maximization in the presene of b oth n ull and soft shaping onstrain ts. The resulting game an b e form ulated as follo ws: ( G 2 ) : maximize Q q 0 R q ( Q q , Q − q ) sub ject to T r G H q Q q G q ≤ P a ve q λ max G H q Q q G q ≤ P peak q U H q Q q = 0 ∀ q = 1 , · · · , Q. (21) W e assume w.l.o.g. that ea h G q is a full-ro w rank matrix, so that the soft shaping onstrain t in ( 21 ) imp oses a onstrain t on the a v erage transmit p o w er radiated b y user q in the whole spae. The soft onstrain ts in (21) are the result of a onstrain t on the o v erall in terferene temp erature limit imp osed b y the primary users [2℄. T ypially , the most stringen t onditions b et w een the p o w er onstrain ts Co.1 and Co.3 is the soft shaping onstrain t Co.3 . This motiv ates the absene in ( 21 ) of the p o w er onstrain t Co.1 , although it ould also b e onsidered. Nash equilibria of game G 2 : W e an deriv e the struture of the Nash equilibria of game G 2 , similarly to what w e did for game G 1 . F or ea h q ∈ Ω , dene the tall matrix U q , G ♯ q U q , where G ♯ q denotes the Monro e-P enrose pseudoin v erse of G q [25 ℄, in tro due the pro jetion matrix P R ( U q ) ⊥ = I − U q ( U H q U q ) − 1 U H q (the orthogonal pro jetion on to R ( U q ) ⊥ ) and the mo died hannel matries H r q = H r q G ♯ H r P R ( U r ) ⊥ , r , q = 1 , · · · , Q. (22) Using the ab o v e denition, w e an no w haraterize the Nash equilibria of game G 2 , as sho wn next. The game G 2 admits a NE, for an y set of hannel matries and n ull/soft shaping onstrain ts. Moreo v er, ev ery NE satises the follo wing set of nonlinear matrix-v alue xed-p oin t equations: Q ⋆ q = G ♯ H q WF q H H q q R − 1 − q ( Q ⋆ − q ) H q q G ♯ q , G ♯ H q V ⋆ q diag p ⋆ q V ⋆H q G ♯ q ∀ q = 1 , · · · , Q, (23) where V ⋆ q = V q ( Q ⋆ − q ) is the semi-unitary matrix with olumns equal to the eigen v etors of matrix H H q q R − 1 − q ( Q ⋆ − q ) H q q , with R − q ( Q − q ) dened in (7), orresp onding to the ¯ L q = rank ( H q q ) p ositiv e eigen- v alues λ ⋆ q ,k = λ q ,k ( Q ⋆ − q ) , and the p o w er allo ation p ⋆ q = p q ( Q ⋆ − q ) satises the follo wing sim ultaneous 12 w aterlling equation: for all k and q , p ⋆ q ( k ) = h µ q − 1 λ ⋆ q,k i P p eak q 0 , if P peak q ¯ L q > P a ve q , P peak q , otherwise, (24) where [ · ] P p eak q 0 denotes the Eulidean pro jetion on to the in terv al [0 , P peak q ] and µ q is hosen to satisfy the p o w er onstrain t P k p ⋆ q ( k ) = P a ve q (see, e.g., [26℄ for pratial algorithms to ompute su h a µ q ). The struture of the Nash equilibria in (23) states that the optimal transmission strategy of ea h user leads to a diagonalizing transmission with a prop er p o w er allo ation, after pre/p ost m ultipliation of the w aterlling solution b y matrix G ♯ q . Similarly to G 1 , the onditions for the uniqueness of the NE of game G 2 an b e obtained, building on the in terpretation of the w aterlling solutions in ( 23) as matrix pro jetion [13℄. As exp eted, the NE of the game is unique, pro vided that the in terferene generated b y seondary users is not to o high. 4 MIMO Asyn hronous Iterativ e W aterlling Algorithm In Setion 3 w e ha v e sho wn that the optimal resoure allo ation among seondary users in hierar hial ognitiv e net w orks orresp onds to an equilibrium of the system, where all the users ha v e maximized their o wn rates, without hamp ering the omm uniations of primary users. Sine there is no reason to exp et a system to b e initially at the equilibrium, the fundamen tal problem b eomes to nd a pro edure that rea hes su h an equilibrium from non-equilibrium states. In this setion, w e fo us on algorithms that on v erge to these equilibria. Sine w e are in terested in a deen tralized implemen tation, where no signaling among seondary and primary users is allo w ed, w e onsider only totally distributed iterativ e algorithms, where ea h user ats indep enden tly of the others to optimize his o wn transmission strategy while p ereiving the other ativ e users as in terferene More sp eially , to rea h the Nash equilibria of the games in tro dued in the previous setion, w e prop ose a fairly general distributed and asyn hronous iterativ e algorithm, alled asyn hronous Iterativ e W aterFilling Algorithm (IWF A). In this algorithm, all seondary users maximize their o wn rate [via the single user MIMO w aterlling solution (12) for game G 1 , (23) for game G 2 , and the lassial MIMO w aterlling solution for games G α and G ∞ ℄ in a total ly asynhr onous w a y , while k eeping the temp erature noise lev els in the liensed bands under the required threshold [2 ℄. A ording to the asyn hronous up dating s hedule, some users are allo w ed to up date their strategy more frequen tly than the others, and they migh t ev en p erform these up dates using outdate d information on the in terferene aused b y the others. Before in tro duing the prop osed asyn hronous MIMO IWF A, w e need the follo wing preliminary de- nitions. W e assume, without loss of generalit y , that the set of times at whi h one or more users up date their strategies is the disrete set T = N + = { 0 , 1 , 2 , . . . } . Let Q ( n ) q denote the o v ariane matrix of the v etor signal transmitted b y user q at the n -th iteration, and let T q ⊆ T denote the set of times n at whi h Q ( n ) q is up dated (th us, at time n / ∈ T q , Q ( n ) q is left un hanged). Let τ q r ( n ) denote the most reen t 13 time at whi h the in terferene from user r is p ereiv ed b y user q at the n -th iteration (observ e that τ q r ( n ) satises 0 ≤ τ q r ( n ) ≤ n ). Hene, if user q up dates his o wn o v ariane matrix at the n -th iteration, then he ho oses his optimal Q ( n ) q , aording to (12) for game G 1 and (23 ) for game G 2 , and using the in terferene lev el aused b y Q ( τ q ( n )) − q , Q ( τ q 1 ( n )) 1 , . . . , Q ( τ q q − 1 ( n )) q − 1 , Q ( τ q q +1 ( n )) q +1 , . . . , Q ( τ q Q ( n )) Q . (25) Some standard onditions in asyn hronous on v ergene theory that are fullled in an y pratial imple- men tation need to b e satised b y the s hedule { τ q r ( n ) } and {T q } ; w e refer to [13℄ for the details. Using the ab o v e notation, the asyn hronous MIMO IWF A is formally desrib ed in Algorithm 1 b elo w, where the mapping in (27 ) is dened as T q ( Q − q ) , ˜ WF q ˜ H H q q R − 1 − q ˜ H q q , q = 1 , · · · , Q , (26) with ˜ WF q ( · ) giv en in (12 ) if the algorithm is applied to game G 1 , and it is dened as T q ( Q − q ) , G ♯ H q WF q H H q q R − 1 − q H q q G ♯ q , q = 1 , · · · , Q , with WF q ( · ) giv en in (23) if the algorithm is applied to game G 2 . The mapping T q ( Q − q ) redues to the lassial MIMO w aterlling solution [19℄ if games G α and G ∞ are onsidered. Algorithm 1: MIMO Asyn hronous IWF A Set n = 0 and Q (0) q = any feasible point ; for n = 0 : N it Q ( n +1) q = T q Q ( τ q ( n )) − q , if n ∈ T q , Q ( n ) q , otherwise ; ∀ q = 1 , · · · , Q (27) end Con v ergene of the asyn hronous IWF A is studied in [ 13 , 14℄ (see also [11 , 12℄ for sp eial ases of the algorithm), where it w as pro v ed that the algorithm on v erges to the NE of the prop osed games under the same onditions guaran teeing the uniqueness of the equilibrium. The prop osed asyn hronous IWF A on tains as sp eial ases a plethora of algorithms, ea h one obtained b y a p ossible hoie of the s hedule { τ q r ( n ) } , {T q } . The sequen tial [2, 11, 27 , 28 ℄ and sim ultaneous [11 ℄-[13℄ IWF As are just t w o examples of the prop osed general framew ork. The imp ortan t result stated in [ 11℄-[13 ℄ is that all the algorithms resulting as sp eial ases of the asyn hronous MIMO IWF A are guaran teed to rea h the unique NE of game under the same set of on v ergene onditions, sine on v ergene onditions do not dep end on the partiular hoie of {T q } and { τ q r ( n ) } [13℄. Moreo v er all the algorithms obtained from Algorithm 1 ha v e the follo wing desired prop erties: - L ow omplexity and distribute d natur e : Ev en in the presene of n ull and/or shaping onstrain ts, the b est resp onse of ea h user q an b e eien tly and lo ally omputed using a MIMO w aterlling based 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 time [iteration index] Rates of secondary users Sequential IWFA Simultaneous IWFA link #6 link #1 link #2 Figure 3: Sim ultaneous vs. sequen tial IWF A: rates of seondary users v ersus iterations, obtained b y the sequen tial IWF A (dashed-line urv es) and sim ultaneous IWF A (solid-line urv es). solution, pro vided that ea h hannel H q q is replaed b y the mo died hannel ˜ H q q (if game G 1 is onsidered) or H q q (if game G 2 is onsidered). Th us, Algorithm 1 an b e implemen ted in a distributed w a y , sine ea h user only needs to measure the o v erall in terferene-plus-noise o v ariane matrix R − q and w aterll o v er ˜ H H q q R − 1 − q ˜ H q q [or o v er H H q q R − 1 − q H q q ℄. - R obustness : Algorithm 1 is robust against missing or outdated up dates of seondary users. This feature strongly relaxes the onstrain ts on the syn hronization of the users' up dates with resp et to those imp osed, for example, b y the sim ultaneous or sequen tial up dating s hemes [11℄-[13 ℄. - F ast onver gen e b ehavior : The sim ultaneous v ersion of the prop osed algorithm on v erges in a v ery few iterations, ev en in net w orks with man y ativ e seondary users. As an example, in Figure 3 w e sho w the rate ev olution of the of 3 links out 8 seondary users, orresp onding to the sequen tial IWF A and sim ultaneous IWF A as a funtion of the iteration index. As exp eted, the sequen tial IWF A is slo w er than the sim ultaneous IWF A, esp eially if the n um b er of ativ e seondary users is large, sine ea h user is fored to w ait for all the users s heduled in adv ane, b efore up dating his o wn o v ariane matrix. This in tuition is formalized in [11 ℄, where the authors pro vided the expression of the asymptoti on v ergen t fator of b oth the sequen tial and sim ultaneous IWF As. - Contr ol of the r adiate d interfer en e : Thanks to the game theoretial form ulation inluding n ull and/or soft shaping onstrain ts, the prop osed asyn hronous IWF A do es not suer of the main dra wba k of the lassial sequen tial IWF A [27℄, i.e., the violation of the in terferene temp erature limits [2 ℄. Figure 4 sho ws an example of the optimal resoure allo ation based on the game theoretial for- m ulation G 1 , for a ognitiv e MIMO net w ork omp osed b y t w o primary users and t w o seondary users, sharing the same sp etrum and spae. Seondary users are equipp ed with four transmit/reeiv e an- tennas, plaed in uniform linear arra ys ritially spaed at half of the w a v elength of the passband transmitted signal. F or the sak e of simpliit y , w e assumed that the hannels b et w een the transmitter and the reeiv er of the seondary users ha v e three ph ysial paths (one line-of-sigh t and t w o reeted paths) as sho wn in Figure 4(a). T o preserv e the QoS of primary users' transmissions, n ull onstrain ts 15 0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180 0 0.5 1 1.5 30 210 60 240 90 270 120 300 150 330 180 0 Null constraint Transmit beamforming pattern of user 1 Transmit beamforming pattern of user 2 (b) Figure 4: Optimal transmit b eamforming patterns at the NE of game G 1 [subplot (b)℄ for a ognitiv e MIMO net w ork omp osed b y t w o primary and t w o seondary users [subplot (a)℄. are imp osed to seondary users in the (line-of-sigh t) diretions of primary users' reeiv ers [see sub- plot (a)℄. F or the senario sho wn in the gure, one n ull onstrain t for ea h pla y er is imp osed along the transmit diretions φ 1 = π / 2 and φ 2 = − 5 π / 12 . This an b e done ho osing for ea h pla y er q the matrix U q in (21) oiniding with the spatial signature v etor in the transmit diretion φ q , i.e., U q = [1 , exp ( − j 2 π ∆ t q sin( φ q )) , exp ( − j 2 π 2∆ t q sin( φ q )) , exp ( − j 2 π 3∆ t q sin( φ q ))] T , with ∆ t q = 1 / 2 denot- ing the normalized (b y the signal w a v elength) transmit an tenna separation and q = 1 , 2 . In Figure 4(b), w e plot the transmit b eamforming patterns, asso iated to the eigen v etors of the optimal o v ariane ma- trix of the t w o seondary users at the NE, obtained using Algorithm 1. In ea h radiation diagram plot, solid (blue) and dashed (bla k) line urv es refer to the t w o eigen v etors orresp onding to the nonzero eigen v alues (arranged in inreasing order) of the optimal o v ariane matrix [reall that, b eause of the n ull onstrain ts, the equiv alen t hannel matrix ˜ H q q in (21 ) has rank equal to 2℄. Observ e that the n ull onstrain ts guaran tee that at the NE no p o w er is radiated b y the t w o seondary transmitters along the diretions φ 1 (for transmitted one) and φ 2 (for transmitted t w o), sho wing that in the MIMO ase, the or- thogonalit y among primary and seondary users an b e rea hed in the spae rather than in the frequeny domain, implying that primary and seondary users ma y share frequeny bands, if this is allo w ed b y F CC sp etrum p oliies. 5 Sp eial Cases The MIMO game theoreti form ulation prop osed in the previous setions pro vides a general and unied framew ork for studying the resoure allo ation problem based on rate maximization in hierar hial CR net w orks, where primary and seondary users o exist. In this setion, w e sp eialize the results to t w o senarios of in terest: 1) the sp etrum sharing problem among primary and seondary users transmitting 16 o v er SISO fr e queny-sele tive hannels ; and 2) the MIMO transeiv ers design of heterogeneous systems sharing the same sp etrum o v er unliensed bands. 5.1 Sp etrum sharing o v er SISO frequeny-seletiv e hannels with sp etral mask onstrain ts The blo k transmission o v er SISO frequeny-seletiv e hannels is obtained from the I/O mo del in (1), when ea h hannel matrix H r q is a N × N T o eplitz irulan t matrix, R n q is a N × N diagonal matrix N is the length of the transmitted blo k (see, e.g., [10℄). This leads to the follo wing eigendeomp osition for ea h hannel H r q = WD r q W H , where W is the normalized IFFT matrix, i.e., [ W ] ij , e j 2 π ( i − 1)( j − 1) / N / √ N for i, j = 1 , . . . , N and D r q is a N × N diagonal matrix, where [ D r q ] k k , H r q ( k ) is the frequeny-resp onse of the hannel b et w een soure r and destination q . Within this setup, w e fo us on game G 1 giv en in (8), but similar results ould b e obtained if game G 2 , G α or G ∞ w ere onsidered instead. In the ase of SISO frequeny-seletiv e hannels, game G 1 an b e rewritten as: maximize Q q 0 log det I + H H q q R − 1 − q H q q Q q sub ject to T r ( Q q ) ≤ P q W H Q q W k k ≤ p max q ( k ) , ∀ k = 1 , · · · , N , ∀ q = 1 , · · · , Q, (28) where { p max q ( k ) } is the set of sp etral mask onstrain ts, that an b e used to imp ose shaping (and th us also n ull) onstrain ts on the transmit p o w er sp etral densit y (PSD) of seondary users o v er liensed/unliensed bands. Nash equilibria: The solutions of the game in (28 ) ha v e the follo wing struture [10 ℄: Q ⋆ q = W Diag ( p ⋆ q ) W H , ∀ q = 1 , · · · , Q, (29) where p ⋆ q , ( p ⋆ q ( k )) N k =1 is the solution to the follo wing set of xed-p oin t equations p ⋆ q = wf q ( p ⋆ − q ) , ∀ q = 1 , · · · , Q, (30) with the w aterlling v etor op erator wf q ( · ) dened as [ wf q ( p − q )] k , " µ q − 1 + P r 6 = q | H r q ( k ) | 2 p r ( k ) | H q q ( k ) | 2 # p max q ( k ) 0 , k = 1 , · · · , N , (31) where µ q is hosen to satisfy the p o w er onstrain t with equalit y P k p ⋆ q ( k ) = P q . Equation (29) states that, in the ase of SISO frequeny-seletiv e hannels, a NE is rea hed using, for ea h user, a m ultiarrier strategy (i.e., the diagonal transmission strategy through the frequeny bins), with a prop er p o w er allo ation. This simpliation with resp et to the general MIMO ase, is a onsequene of the prop ert y that all hannel T o eplitz irulan t matries are diagonalized b y the same matrix, i.e., the IFFT matrix W , that do es not dep end on the hannel realization. 17 In terestingly , m ultiarrier transmission with a prop er p o w er allo ation for ea h user is still the opti- mal transmission strategy if in (28 ) instead of the information rate, one onsiders the maximization of the transmission rate using nite order onstellations and under the same onstrain ts as in (28) plus a onstrain t on the a v erage error probabilit y . Using the gap appro ximation analysis, the optimal p o w er allo ation is still giv en b y the w aterlling solution (31), where ea h hannel transfer funtion | H q q ( k ) | 2 is replaed b y | H q q ( k ) | 2 / Γ q , where Γ q ≥ 1 is the gap [10℄. The gap dep ends only on the family of onstella- tion and on error probabilit y onstrain t P e,q ; for M -QAM onstellations, for example, the resulting gap is Γ q = ( Q − 1 ( P e,q / 4)) 2 / 3 (see, e.g., [29 ℄). Rea hing a NE of the game in (28 ) satises a omp etitiv e optimalit y priniple, but, in general, m ultiple equilibria ma y exist, so that one is nev er sure ab out whi h equilibrium is really rea hed. Suien t onditions on the MUI that guaran tee the uniqueness of the equilibrium ha v e b een prop osed in the literature [10 ℄-[12 ℄ and [27, 28℄. Among all, one of the t w o follo wing onditions is suien t for the uniqueness of the NE: X r 6 = q max k ¯ H r q ( k ) 2 ¯ H q q ( k ) 2 d 2 q q d 2 r q < 1 , ∀ q = 1 , · · · , Q, and ∀ k = 1 , · · · , N , (32) X r 6 = q max k ¯ H r q ( k ) 2 ¯ H q q ( k ) 2 d 2 q q d 2 r q < 1 , ∀ r = 1 , · · · , Q , and ∀ k = 1 , · · · , N , (33) where w e ha v e in tro dued the normalized hannel transfer funtions H r q ( k ) , ¯ H r q ( k ) /d 2 r q , ∀ r , q , with d r q indiating the distane b et w een transmitter of the r -th link and the reeiv er of the q -th link. F rom ( 32)- (33), it follo ws that, as exp eted, the uniqueness of NE is ensured if seondary users are suien tly far apart from ea h other. In fat, from (32)-(33 ) for example, one infers that there exists a minim um distane b ey ond whi h the uniqueness of NE is guaran teed, orresp onding to the maxim um lev el of in terferene that ma y b e tolerated b y the users. Sp eially , ondition (32 ) imp oses a onstrain t on the maxim um amoun t of in terferene that ea h reeiv er an tolerate; whereas ( 33 ) in tro dues an upp er b ound on the maxim um lev el of in terferene that ea h transmitter is allo w ed to generate. In terestingly , the uniqueness of the equilibrium do es not dep end on the in terferene generated b y the transmissions of primary users. Asyn hronous IWF A: T o rea h the equilibrium of the game, seondary users an p erform the asyn- hronous IWF A based on the mapping in (31 ). This algorithm an b e obtained diretly from Algorithm 1, as sp eial ase. It w as pro v ed in [12 ℄ that, e.g., under onditions (32 )-(33), the asyn hronous IWF A based on mapping (31) on v erges to the unique NE of game in (28) as Nit → ∞ , for an y set of feasible initial onditions and up dating s hedule. In Figure 5, w e sho w an example of the optimal p o w er allo ation in SISO frequeny-seletiv e hannels at the NE, obtained using the prop osed asyn hronous IWF A, for a CR system omp osed b y one primary user [subplot (a)℄ and t w o seondary users [subplot (b)℄, sub jet to n ull onstrain ts o v er liensed bands, sp etral mask onstrain ts and transmit p o w er onstrain ts. In ea h plot, solid and dashed-dot line urv es refer to optimal PSD of ea h link and PSD of the MUI plus thermal noise, normalized b y the hannel 18 50 100 150 200 250 300 350 400 450 500 10 −3 10 −2 10 −1 10 0 10 1 frequency Optimal PSD of Primary User Licensed band 0 50 100 150 200 250 300 350 400 450 500 10 −4 10 −2 10 0 frequency Optimal PSD of Secondary User 1 0 50 100 150 200 250 300 350 400 450 500 10 −3 10 −2 10 −1 10 0 10 1 frequency Optimal PSD of Secondary User 2 Spectral Mask Null Constraints (licensed band) Null Constraints (licensed band) Spectral Mask (a) (b) Figure 5: Sp etrum sharing among one primary [subplot (a)℄ and t w o seondary users [subplot (b)℄: Optimal PSD of ea h link (solid lines), and PSD of the MUI-plus-thermal noise normalized b y the hannel transfer funtion square mo dulus of the link (dashed-dot line). transfer funtion square mo dulus of the link, resp etiv ely . In this example, there is a band A (from 50 to 300 frequeny bins) allo ated to an ativ e primary user; there is then a band B (from 300 to 400 frequeny bins) allo ated to liensed users, but temp orarily un used; the rest of the sp etrum, denoted as C , is v aan t. The temp orarily v oid band B an b e utilized b y seondary users, pro vided that they do not o v erome a maxim um tolerable sp etral densit y . The optimal p o w er allo ations sho wn in Figure 5 are the result of running the sim ultaneous IWF A. W e an observ e that the seondary users do not transmit o v er band A and they allo ate their p o w er o v er b oth bands B and C , resp eting a p o w er sp etral densit y limitation o v er band B . 5.2 MIMO transeiv er design of heterogeneous systems in unliensed bands W e onsider no w on a senario where m ultiple unliensed MIMO ognitiv e users share the same unliensed sp etrum and geographial area. The a v ailabilit y of MIMO transeiv ers learly enri hes the p ossibilities for sp etrum sharing as it adds the extra spatial degrees of freedom. In unliensed bands, there are no in terferene onstrain ts to b e satised b y the users. Th us, the game theoretial form ulation as giv en in (8), without onsidering the n ull onstrain ts, seems the most appropriate to study the resoure allo ation problem in this senario. In the follo wing w e refer to game G 1 assuming taitly that the n ull onstrain ts are remo v ed. Similarly to the SISO ase, suien t onditions for the uniqueness of the NE are giv en b y one of the t w o follo wing set of onditions (more general onditions are giv en in [13℄): Lo w MUI reeiv ed: X r 6 = q ρ H H r q H − H q q H − 1 q q H r q < 1 , ∀ q = 1 , · · · , Q, (34) Lo w MUI generated: X q 6 = r ρ H H r q H − H q q H − 1 q q H r q < 1 , ∀ r = 1 , · · · , Q. (35) Conditions (34 )-(35 ) quan tify ho w m u h MUI an b e tolerated b y the systems to guaran tee the uniqueness of the NE. In terestingly , (32 )-(33 ) and most of the onditions kno wn in the literature [11 , 27 , 28 ℄ for the 19 uniqueness of the NE of the rate-maximization game in SISO frequeny-seletiv e in terferene hannels and OFDM transmission ome naturally from (34 )-(35 ) as sp eial ases. The Nash equilibria of game G 1 an b e rea hed using the asyn hronous IWF A desrib ed in Algorithm 1, whose on v ergene is guaran teed under onditions (34 )-(35), for an y set of initial onditions and up dating s hedule of the users. 1 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 16 18 Users’ Sum−Rate n T =n R =6 n T =n R =4 n T =n R =2 n T =n R =1 d rq / d qq Figure 6: Sum-Rate of the users v ersus the in ter-pair distane d r q /d qq ; d r q = d qr , d r r = d qq = 1 , ∀ r, q , for dieren t n um b ers of transmit/reeiv e an tennas . In Figure 6 w e sho w an example of the b enets of MIMO transeiv ers in the ognitiv e radio on text. W e plot in the gure the sum-rate of a t w o-user frequeny-seletiv e MIMO system as a funtion of the in ter-pair distane among the links, for dieren t n um b er of transmit/reeiv e an tennas. The rate urv es are a v eraged o v er 500 indep enden t hannel realizations, whose taps are sim ulated as i.i.d. Gaussian random v ariables with zero mean and unit v ariane. F or the sak e of simpliit y , the system is assumed to b e symmetri, i.e., the transmitters ha v e the same p o w er budget and the in terferene links are at the same distane (i.e., d r q = d q r , ∀ q , r ), so that the ross hannel gains are omparable in a v erage sense. F rom the gure one infers that, as for isolated single-user systems or m ultiple aess/broadast hannels, also in MIMO in terferene hannels, inreasing the n um b er of an tennas at b oth the transmitter and the reeiv er side leads to a b etter p erformane. The in teresting result, oming from Figure 6, is that the inremen tal gain due to the use of m ultiple transmit/reeiv e an tennas is almost indep enden t of the in terferene lev el in the system, sine the MIMO (inremen tal) gains in the high-in terferene ase (small v alues of d r q /d q q ) almost oinide with the orresp onding (inremen tal) gains obtained in the lo w-in terferene ase (large v alues of d r q /d q q ), at least for the system sim ulated in Figure 6. This desired prop ert y is due to the fat that the MIMO hannel pro vides more degrees of freedom for ea h user than those a v ailable in the SISO hannel, that an b e explored to nd out the b est partition of the a v ailable resoures for ea h user, p ossibly anelling the MUI. 20 6 Conlusion and Diretions for F urther Dev elopmen ts In this pap er w e ha v e prop osed a signal pro essing approa h to the design of CR systems, using a omp etitiv e optimalit y priniple, based on game theory . W e ha v e addressed and solv ed some of the hallenging issues in CR, namely: 1) the establishmen t of onditions guaran teeing that the dynamial in teration among ognitiv e no des, under onstrain ts on the transmit sp etral mask and on in terferene indued to primary users, admits a (p ossibly unique) equilibrium; and 2) the design of deen tralized algorithms able to rea h the equilibrium p oin ts, with minimal o ordination among the no des. W e ha v e seen ho w basi signal pro essing to ols su h as subspae pro jetors pla y a fundamen tal role. The sp etral mask onstrain ts ha v e b een in fat used in a v ery broad sense, meaning that the pro jetion of the transmitted signal along presrib ed subspaes should b e n ull (n ull onstrain ts) or b elo w a giv en threshold (soft onstrain ts). The on v en tional sp etral mask onstrain ts an b e seen as a simple ase of this general set-up, v alid for SISO hannels and using as subspaes the spae spanned b y the IFFT v etors with frequenies falling in the guard bands. This general setup enompasses m ultian tenna MIMO systems, whi h is partiularly useful for CR, as it pro vides the additional spatial degrees of freedom to on trol the in terferene generated b y the ognitiv e users. Of ourse, this eld of resear h is full of in teresting further diretions w orth of in v estigation. The NE p oin ts deriv ed in this pap er w ere ditated b y the need of nding totally deen tralized algorithms with minimal o ordination among the no des. Ho w ev er, the NE p oin ts ma y not b e P areto-eien t. This raises the issue of ho w to mo v e from the NE to w ards the P areto optimal trade-o surfae, still using a deen tralized approa h. Game theory itself pro vides a series of strategies to mo v e from ineien t Nash equilibria to w ards P areto-eien t solutions, still using a deen tralized approa h, through, for example, rep eated games, where the pla y ers learn from their past hoies [9℄. Examples of su h games are the aution games, where the autioneer (primary users) dynamially determine resoure allo ation and pries for the bidders (seondary users), dep ending on tra demands, QoS and supply/demand urv es, as evidened in a series of w orks (see, e.g., [30, 31 , 32℄). Rep eated games ma y also tak e the form of negotiations b et w een primary and seondary users, with primary users willing to lease part of their sp etrum to seondary users, under suitable rem unerations [16℄ or under the a v ailabilit y giv en b y seondary users to establish o op erativ e links with the primary users to impro v e their QoS [33℄. Comp etitiv e priing for sp etrum sharing w as also prop osed as an oligop oly mark et where a few primary users oer sp etrum aess opp ortunities to seondary users [34 ℄. An in teresting issue will b e the in tegration of our asyn hronous IWF A in rep eated (aution) games, where the optimization onsiders a set of primary users oering the lease of p ortion of their resoures to a set of seondary users, as a funtion of tra demands, QoS requiremen ts and ph ysial onstrain ts. Our sear h for the uniqueness onditions of the NE and the on v ergene onditions of our prop osed algorithms fored us to simplify the mo del. F or example, w e assumed that ea h reeiv er has an error-free short-term predition of the hannel. This assumption w as neessary for the mathematial tratabilit y 21 of the problem and to b e able to pro vide losed-form expressions of our ndings. This is useful to gain a full understanding of the problem, without relying on sim ulation results only . Ho w ev er, in pratie, the transmitter is only able to aquire an estimate aeted b y errors and, based on that, to form a predition of the short term future ev olution. An in teresting extension of the presen ted approa h onsists then in taking in to aoun t the eets of estimation errors and dev eloping robust strategies. This is partiularly relev an t in CR systems b eause the strategy adopted b y the ognitiv e users ma y b e more or less aggressiv e dep ending on the reliabilit y of their hannel sensing. Channel iden tiation has a long history in signal pro essing. The problem b eomes esp eially hal- lenging in CR net w orks, where the estimation of the hannel v oids, for example, m ust b e v ery aurate. Nev ertheless, the estimation itself ma y b e impro v ed b y exploiting the a v ailabilit y of a net w ork of no des that ould, in priniple, o op erate to get b etter and b etter estimates of the eletromagneti en vironmen t, w orking as a sensor net w ork of ognitiv e no des. Referenes [1℄ F CC Sp etrum P oliy T ask F ore, F CC Rep ort of the Sp etrum Eieny W orking Group, No v. 2002. http://www.f.gov/sptf/files/S EWGFi nalRe port 1.pdf . [2℄ S. Ha ykin, Cognitiv e Radio: Brain-Emp o w ered Wireless Comm uniations, IEEE Jour. on Sele te d A r e as in Commu- ni ations , v ol. 23, no. 2, pp. 201-220, F ebruary 2005. [3℄ I. F. Akyildiz, W.-Y. Lee, M. C. V uran, S. Mohan t y , NeXt Generation/Dynami Sp etrum A ess/Cognitiv e Radio Wireless Net w orks: A Surv ey , Computer Networks , v ol. 50, pp. 21272159, 2006. [4℄ Q. Zhao and B. Sadler, A Surv ey of Dynami Sp etrum A ess, IEEE Signal Pr o essing Magazine, v ol. 24, no. 3, pp. 7989, Ma y 2007. [5℄ A. Goldsmith, S. A. Jafar, I. Mari, S. Sriniv asa, Breaking Sp etrum Gridlo k with Cognitiv e Radios: An Information Theoreti P ersp etiv e, to app ear on IEEE Jour. of Sele te d A r e as in Communi ations (JSA C ). [6℄ J. Mitola, Cognitiv e Radio for Flexible Mobile Multimedia Comm uniation, in Pro . of the IEEE International W orkshop on Mobile Multime dia Communi ations (MoMuC) 1999 , pp. 310, No v em b er 1999. [7℄ N. Devro y e, P . Mitran, V. T arokh, A hiev able Rates in Cognitiv e Radio Channels, IEEE T r ans. on Information The ory , v ol. 52, pp. 18131827, Ma y 2006. [8℄ M. J. Osb orne and A. Rubinstein, A Course in Game The ory , MIT Press, 1994. [9℄ T. Basar and G. J. Olsder, Dynami Non o op er ative Game The ory , SIAM Series in Classis in Applied Mathematis, Philadelphia, Jan uary 1999. [10℄ G. Sutari, D. P . P alomar, and S. Barbarossa, Optimal Linear Preo ding Strategies for Wideband Non-Co op erativ e Systems based on Game Theory-P art I: Nash Equilibria, IEEE T r ans. on Signal Pr o essing, v ol. 56, no. 3, pp. 12301249, Mar h 2008. [11℄ G. Sutari, D. P . P alomar, and S. Barbarossa, Optimal Linear Preo ding Strategies for Wideband Non-Co op erativ e Systems based on Game Theory-P art I I: Algorithms, IEEE T r ans. on Signal Pr o essing, v ol. 56, no. 3, pp. 12501267, Mar h 2008. [12℄ G. Sutari, D. P . P alomar, and S. Barbarossa, Asyn hronous Iterativ e W aterlling for Gaussian F requeny-Seletiv e In terferene Channels, in IEEE T r ans. on Information The ory , v ol. 54, no. 7, pp. 28682878, July 2008. [13℄ G. Sutari, D. P . P alomar, and S. Barbarossa, Comp etitiv e Design of Multiuser MIMO Systems Based on Game Theory: A Unied View, IEEE Jour. of Sele te d A r e as in Communi ations (JSA C ), sp eial issue on Game Theory in Comm uniation Systems , v ol. 26, no.7, Septem b er 2008. [14℄ G. Sutari, D. P . P alomar, and S. Barbarossa, MIMO Cognitiv e Radio: A Game Theoretial Approa h,submitted to IEEE T rans. on Signal Pro essing (2008). See also Pro . of the 9th IEEE W orkshop on Signal Pr o essing A dvan es for Wir eless Communi ations ( SP A W C 08 ), Reife, P ernam buo, Brazil, July 6-9, 2008. 22 [15℄ Q. Zhao, Sp etrum Opp ortunit y and In terferene Constrain t in Opp ortunisti Sp etrum A ess, in Pro . of IEEE International Confer en e on A oustis, Sp e e h, and Signal Pr o essing (ICASSP), Honolulu, Ha w aii, USA, April 15-20, 2007. [16℄ L. Cao, H. Zheng, Distributed Sp etrum Allo ation via Lo al Bargaining, in Pro . of IEEE SECON 2005 , pp. 475 486, Sept. 2005. [17℄ Y. Chen, Q. Zhao, A. Sw ami, Join t design and separation priniple for opp ortunisti sp etrum aess in the presene of sensing errors, IEEE T r ans. on Information The ory , v ol. 54, pp. 2053 - 2071, Ma y 2008. [18℄ B. Wild and K. Ram handran, Deteting primary reeiv ers for ognitiv e radio appliations,in Pro . of the IEEE Symp. New F r ontiers Dynami Sp e trum A ess Networks ( D YSP AN 05 ), pp. 124130, No v. 2005. [19℄ T. M. Co v er and J. A. Thomas, Elements of Information The ory , John Wiley and Sons, 1991. [20℄ W. Y u and R. Lui, Dual metho ds for nonon v ex sp etrum optimization of m ultiarrier systems, IEEE T r ans. Com- mun ., v ol. 54, pp. 13101322, 2006. [21℄ R. Cendrillon, W. Y u, M. Mo onen, J. V erliden, and T. Bosto en, Optimal m ulti-user sp etrum managemen t for digital subsrib er lines, IEEE T r ans. Commun., v ol. 54, no. 5, pp. 922933, Ma y 2006. [22℄ S. Y e and R. S. Blum, Optimized Signaling for MIMO In terferene Systems with F eedba k, IEEE T r ans. on Signal Pr o essing , v ol. 51, no. 11, pp. 2839-2848, No v em b er 2003. [23℄ Y. Rong and Y. Hua, Optimal p o w er s hedule for distributed MIMO links, in Pro . of the 25th A rmy Sien e Confer en e, No v em b er 2006. [24℄ Z.-Q. Luo and S. Zhang, Dynami Sp etrum Managemen t: Complexit y and Dualit y , IEEE Jour. of Sele te d T opis in Signal Pr o essing , v ol. 2, no. 1, pp. 5772, F ebruary 2008. [25℄ R. A. Horn and C. R. Johnson, Matrix A nalysis , Cam bridge Univ. Press, 1985. [26℄ Daniel P . P alomar and J. F onollosa, Pratial Algorithms for a F amily of W aterlling Solutions, IEEE T r ans. on Signal Pr o essing , v ol. 53, no. 2, pp. 686695, F eb. 2005. [27℄ W. Y u, G. Ginis, and J. M. Cio, Distributed Multiuser P o w er Con trol for Digital Subsrib er Lines, IEEE JSA C , v ol. 20, no. 5, pp. 1105-1115, June 2002. [28℄ Z.-Q. Luo and J.-S. P ang, Analysis of Iterativ e W aterlling Algorithm for Multiuser P o w er Con trol in Digital Sub- srib er Lines," EURASIP Jour. on Applie d Signal Pr o essing , Ma y 2006. [29℄ J. G. D. F orney and M. V. Eyub oglu, Com bined Equalization and Co ding Using Preo ding, IEEE Communi ations Magazine , v ol. 29, no. 12, pp. 2534, Deem b er 1991. [30℄ Y.-C. Liang, H.-H. Chen, J. Mitola, P . Mahonen, R. K ohno, J. H. Reed Eds., Sp eial Issue on Cognitiv e Radio: Theory and Appliation, IEEE Jour. of Sele te d A r e as in Communi ations (JSA C ), v ol. 26, no. 1, Jan. 2008. [31℄ A. Sw ami, R. A. Berry , A. M. Sa y eed, V. T arokh, Q. Zhao Eds., Sp eial Issue on Signal Pro essing and Net w orking for Dynami Sp etrum A ess, IEEE Jour. of Sele te d T opis in Signal Pr o essing , v ol. 2, no. 1, F eb. 2008. [32℄ Z. Ji, K. Ra y Liu, Dynami Sp etrum Sharing: A Game Theoretial Ov erview, IEEE Communi ations Ma g azin e, v ol. 45, no. 5, pp. 8894, Ma y 2007. [33℄ O. Simeone, I. Stano jev, S. Sa v azzi, Y. Bar-Ness, U. Spagnolini, R. Pi kholtz, Sp etrum leasing to o op erating ad ho seondary net w orks, IEEE Jour. of Sele te d A r e as in Communi ations (JSA C ), v ol. 26, no. 1, pp. 203213, Jan. 2008. [34℄ D. Niy ato, E. Hossain, Comp etitiv e Priing for Sp etrum Sharing in Cognitiv e Radio Net w orks: Dynami Game, Ineieny of Nash Equilibrium, and Collusion, IEEE Jour. of Sele te d A r e as in Communi ations (JSA C ), v ol. 26, no. 1, pp. 192202, Jan. 2008. 23
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