Distributed Interference Alignment in Cognitive Radio Networks

Transcription

1 Dstrbuted Interference Algnment n Cogntve Rado Networks Y Xu and Shwen Mao Department of Electrcal and Computer Engneerng, Auburn Unversty, Auburn, AL, USA Abstract In ths paper, we nvestgate the problem of ncorporatng two advanced physcal layer technologes,.e., multplenput and multple-output (MIMO) and dstrbuted nterference algnment, n cogntve rado (CR) networks. We present a cooperatve spectrum leasng scheme for prmary and secondary users to trade off between data transmsson and revenue collecton/payment. A Stackelberg game s formulated, where the prmary user s the leader and the secondary users are followers. Wth backward nducton, we derve the unque Stackelberg Equlbrum, where no player can gan by unlaterally changng strategy, as well as the optmal strateges. We fnd spectrum leasng s always benefcal to enhance the utltes of prmary and secondary users. The proposed scheme outperforms a nospectrum-leasng scheme and a cooperatve scheme presented n the lterature wth consderable gans, whch demonstrate the benefts of spectrum leasng and dstrbuted nterference algnment and valdate the effcacy of the proposed scheme. I. INTRODUCTION Due to the tremendous ncrease n wreless data, rado spectrum s quckly depleted. However, accordng to the FCC report [1], whle some lcensed bands are overcrowded, many others are underutlzed. Under tradtonal fxed spectrum allocaton polcy, when lcensed users (or, prmary users) are not actve, the channels assgned to them are wasted (termed as spectrum opportuntes). Cogntve rados (CR) are proposed as a new wreless paradgm for explotng such spectrum opportuntes, to enable flexble and effcent access to rado spectrum. In CR networks, unlcensed users (or, secondary users) can access the lcensed band opportunstcally, whle prmary users gan by collectng revenue for spectrum leasng. Such a CR paradgm has been shown to have hgh potentals to enhance spectrum effcency []. As sgnfcant advances are made n many aspects of CR research, such as spectrum sensng and dynamc spectrum access, t s also desrable to ncorporate advanced physcal layer technques nto CR networks. One of such technques s Multple-Input and Multple- Output (MIMO), whch can be used to reduce bt error rate, transmt more packets, or strengthen the sgnal to nterference and nose rato (SINR). In the past decade, MIMO has evolved from a theoretc concept to a technology that can be wdely used n practce [3]. It s desrable to explot MIMO for enhanced prmary and secondary transmssons. The second physcal layer technology s nterference algnment, a sgnfcant breakthrough that explots nterference n nterference lmted wreless networks [4]. Tradtonally, f nterference s small, t s smply treated as background nose; f nterference s large, t can be decoded frst and then removed from the receved sgnal (.e., nterference cancellaton); f nterference s comparable to the desred sgnal, we usually try to avod ths case by orthogonalzng the channels or adopt a medum access control (MAC) scheme [5], [6]. Unlke tradtonal approaches, nterference algnment casts nterference to half of the receved sgnal space to acheve a normalzed Degree of Freedom (DoF) of K/, where K s the number of nterferng users. Snce an nterference-free channel only has a normalzed DoF of 1, substantal system throughput gan can be acheved wth nterference algnment when K s large. For nterference algnment, a strong requrement s the avalablty of global channel state nformaton (CSI) at every node. To relax ths requrement, dstrbuted nterference algnment s nvestgated and an teratve algorthm s proposed n [7] to acheve nterference algnment wth local CSI. In ths paper, we nvestgate how to ncorporate these two advanced physcal layer technologes,.e., MIMO and dstrbuted nterference algnment, n CR networks. The CR network conssts of a prmary user and multple secondary users, each wth N antennas. Tme s dvded nto equal length tme slots wth a normalzed length. The prmary user has some data to send and requres a certan non-zero data rate n each tme slot. It also leases spectrum to secondary users for more revenue. Secondary users pay the prmary user for data transmsson n the tme slot. In the proposed cooperatve spectrum leasng scheme, the prmary user dvdes the tme slot nto three phases: () n Phase I, only the prmary user transmts wth MIMO; () n Phase II, the prmary user and a selected set of secondary users transmt smultaneously usng dstrbuted nterference algnment; () n Phase III, only selected secondary users transmt wth dstrbuted nterference algnment. The prmary user decdes the dvson of the three phases, selects the set of secondary users for spectrum leasng, and collects a revenue from the selected secondary users proportonal to ther transmt powers (or, data rates). We fnd such a cooperatve spectrum leasng framework fts well wth the Stackelberg game theory [8]. In the formulated Stackelberg game, the prmary user s the leader and the secondary users are followers. The leader decdes the dvson of a tme slot nto three phases and selecton of followers, amng to balance ts own data transmsson and revenue collecton by leasng spectrum. Once the leader decsons are made, a follower can choose a transmt power (and the correspondng data rate) based on how much t s wllng to pay. We defne the Stackelberg Equlbrum where nether the prmary user nor any secondary user could gan by unlateral change of strategy.

2 We present a rgorous analyss wth the backward nducton method [8] and derve the unque Stackelberg Equlbrum for the cooperatve spectrum leasng game. We fnd the most desrable scenaro for secondary users s to have only Phase III n the tme slot wth only 3 players. The strategy for the prmary user depends on the number of secondary users. Wth more than N secondary users, exactly N secondary users wll be selected, each havng one nterference free channel, and there wll be only Phase II n the tme slot. Wth fewer than N secondary users, all secondary users wll be selected and there wll be only Phases II and III n the tme slot. Therefore, spectrum leasng s always helpful for ncreasng the utltes of both the prmary and secondary users. In the smulaton study, we frst compare the proposed scheme wth a scheme wthout spectrum leasng to demonstrate the benefts of spectrum leasng. We then compare the proposed scheme wth the cooperatve scheme presented n [9] to demonstrate the effcacy of dstrbuted nterference algnment. Sgnfcant performance gans are acheved by the proposed scheme n these smulatons. The remander of ths paper s organzed as follows. In Secton II, we present the prelmnares and system model. We defne the Stackelberg game n Secton III and derve the unque Stackelberg equlbrum n Secton IV. Smulaton results are presented n Secton V and related work s revewed n Secton VI. Secton VII concludes ths paper. II. PRELIMINARIES AND SYSTEM MODEL A. MIMO and Dstrbuted Interference Algnment Ths paper s closely related to MIMO and dstrbuted nterference algnment. We bref revew the prelmnares n ths secton. More detals can be found n [3], [7]. 1) MIMO Capacty Bascs: Wth the advance of antenna technology, t s now feasble to equp wreless devces wth multple antennas. In general, three types of performance gans can be acheved wth MIMO, namely, dversty gan, multplexng gan, and antenna gan. In ths paper, we focus on multplexng gan, namely, DoF. We assume that all transmtters and recevers have the same number of antennas. For a MIMO system wth N antennas, assume that the CSI H s known at the transmtter. Snce the MIMO channel can be decomposed nto d parallel channels, the channel capacty s gven by [1] C = max p : p P d =1 ( log 1+ σ p ), (1) where P denotes the total transmtter power lmt, p s the power allocated to the -th parallel channel, σ = λ and λ s the -th largest egenvalue of matrx HH H. Note that bandwdth s normalzed throughout ths paper. In the hgh SNR regon, equal power allocaton s shown to be sub-optmal, but s easer for mathematcal modelng than water-fllng. When the transmt power s P/d for each parallel PU_t SU1_t SU_t SU3_t PU_r SU1_r SU_r SU3_r PU_t SU1_t SU_t SU3_t PU_r SU1_r SU_r SU3_r PU_t SU1_t SU_t SU3_t PHASE I PHASE II PHASE III αβ α(1 β ) (1 α) PU_r SU1_r SU_r SU3_r Fg. 1. The three-phase operaton of the MIMO CR network wth dstrbuted nterference algnment. channel, the total capacty can be approxmated as d ( ) d ( ) C log 1+ Pσ Pσ log dn =1 dn =1 d ( ) σ = d log(snr)+ log. () d =1 The second tem n () s neglgble when the SNR s hgh. We thus gnore ths term n the followng sectons. ) Dstrbuted Interference Algnment: The basc dea of nterference algnment s to cast the nterference to no more than half of the receved sgnal space, and leave the other half clean and recognzable. If there are K users, totally K/ normalzed DoF could be acheved. The system throughput can be greatly enhanced when K s large. For K =and 1, there s no nterference; for K =, the normalzed DoF s 1, whch s trval. Therefore, we only consder the case where the number of nterferng nodes K satsfes K 3. It s worth notng that, to algn nterference perfectly, global CSI s requred at every partcpatng node. To overcome ths challenge, an teratve dstrbuted nterference algnment algorthm was proposed n [7], whch only requres local CSI at each nterferng node. In [11], a system s sad to be proper f t satsfes d N K+1. We consder a proper system to be feasble for dstrbuted nterference algnment for smplcty. B. System Model and Assumptons The MIMO CR network s llustrated n Fg. 1. There are one prmary user and K T secondary users sharng the lcensed spectrum, each wth N antennas. We consder a tme-slotted system, where each tme slot s normalzed to 1 unt n length and s dvded nto three phases, wth lengths αβ, α(1 β), and (1 α), respectvely, for fractons α 1 and β 1. In Phase I, the prmary user transmts ts packets at the hghest rate usng MIMO, and all secondary users reman slent. The DoF for the prmary user s d I = N. The achevable rate of the prmary user n Phase I s: RP I = d I log(snr), (3)

3 where SNR s assumed to be constant durng a tme slot. We assume that the prmary user always has a fnte amount of packets to send n each tme slot. After a perod of hgh data rate transmsson (wth length αβ), the prmary user has the ncentve to lease the spectrum to secondary users to ncrease ts utlty, by collectng revenue from selected secondary users (but at the cost of a lower data rate for tself). In Phase II, the prmary user and K [,K T ] selected secondary users transmt smultaneously usng dstrbuted nterference algnment, wth a DoF of d II = N K+. A selected secondary user makes payments that s proportonal to ts transmt power (.e., ts data rate), and the prmary user collects payments from all selected secondary users. The achevable rate of the prmary user n Phase II s RP II = d II log(snr). (4) The achevable rate of secondary user S n Phase II s RS II = d II log(snr ), (5) where SNR = P / s the SNR for each selected secondary user, whch s assumed to be constant n a tme slot. In Phase III, the prmary user stops ts transmsson and leases the spectrum to selected secondary users, whch transmt usng dstrbuted nterference algnment wth d III = In Phase III, the achevable rate of secondary user S s N K+1 RS III = d III log(snr ). (6) We assume a common control channel for the prmary user and secondary users to exchange precodng and nterference cancellaton matrces, the weght factor nformaton, and the fractons α and β. Channel estmaton s completed before data transmssons. Note that the DoFs are ntegers. III. STACKELBERG GAME FORMULATION In the MIMO CR network, the prmary user decdes the dvson of a tme slot nto three phases and selecton of secondary users, whle balancng ts own data transmsson and revenue collecton by leasng spectrum. Once the decsons are made by the prmary user, a secondary user can choose a transmt power (and the correspondng data rate) based on how much t s wllng to pay. Such nteractons ft perfectly wth the Stackelberg game model [8]. In ths secton, we formulate a Stackelberg game for the MIMO CR network wth dstrbuted nterference algnment. The prmary user s the leader and the secondary users are followers. Thestrategy of the prmary user s gven by S P = {α, β, K α 1, β 1, 3 K K T }. (7) The secondary user strategy s to fnd a transmt power P,as S S = {P P P max },. (8) Here we assume that P max w S N/C, where C s unt prce for secondary user transmt power (see (9)) and w S s the weght factor for secondary user utlty (see (1)).. The prmary user transmts ts data n Phases I and II, and collects revenue n Phases II and III. The utlty of the prmary user s the sum of data transmtted and revenue collected, as K U P = w P f P (R P )+ C P, (9) k=1 where R P = αβrp I + α(1 β)rii P s the amount of prmary user data transmtted, w P s a weght factor, C s the unt prce for secondary user power, and f P (x) s the satsfacton functon of the prmary user. Snce the prmary user always has some data to send, t requres a mnmum data rate. Naturally we choose f P (x) =ln(x), x. The negatve value for very small x serves as a penalty that forces the prmary user to acheve a mnmum data rate. From the shape of f P (x), the prmary user s enthusastc about data transmsson at the begnnng stage. After a perod of hgh rate transmsson, even a great ncrease n data transmsson can only result n a small ncrease n ts satsfacton. Snce the prmary user s ratonal and selfsh, t ams to maxmze U P by controllng the lengths of the three phases and selectng secondary users to partcpate n the game. By adjustng weght w P, the prmary users can trade off between data transmsson and revenue collecton. Selected secondary users transmt ther data durng Phases II and III and make a one-tme payment to the prmary user. The utlty of a secondary user S s gven by U S = w S f S (R S ) C P. (1) where R S = α(1 β)rs II +(1 α)rs III s the amount of data S transmts, f S (x) s the secondary user satsfacton functon, and w S s the weght factor. To smplfy notaton, we assume dentcal w S for all secondary users. Snce the essence of CR s to opportunstcally explot spectrum, we choose f S (x) =x, ndcatng that the secondary users operate n the best-effort manner. The weght w S allows a secondary user to trade off between data transmsson and payment. Therefore we defne a Stackelberg game, wth players, ther roles, strateges ((7) and (8)), and utltes ((9) and (1)) specfed. We provde a thorough analyss of the game wth respect to the exstence and unqueness of the Stackelberg Equlbrum and optmal strateges n Secton IV. IV. PERFORMANCE ANALYSIS AND SOLUTION STRATEGY Let P be the vector of secondary user powers and P = P \P. We frst defne Stackelberg Equlbrum as follows. Defnton 1. (Stackelberg Equlbrum) A strategy set {α, β, K, P } s a Stackelberg Equlbrum of the game defned n Secton III f the followng condtons are satsfed: 1) U P (α,β,k, P ) U P (α, β, K, P ), for all α [, 1], β [, 1], and K [,K T ]. ) U S (P, P,α,β,K ) U S (P, P,α,β,K ), for all α [, 1], β [, 1], K [,K T ] and [1,K]. Usng the backward nducton method [8], we prove the unqueness of the Stackelberg Equlbrum, and derve the unque Stackelberg Equlbrum (and the optmal strategy) for the game defned n Secton III n the remander of ths secton.

4 A. Secondary User Utlty Maxmzaton From (1), the utlty of secondary user S s gven by U S (P ) = w S [α(1 β)d II log(p / ) + (1 α)d III log(p / )] C P. To maxmze ts utlty, the secondary user solves the followng maxmzaton problem. max U S (P ). (11) P P max For gven α and β, U S (P ) s a concave functon of P. Settng du S /dp =, we derve the unque maxmzer of (11), as P =[w S α(1 β)d II + w S (1 α)d III ]/C. (1) Snce α 1, β 1 and d II d III N, wehave P w S d III /C w S N/C P max, ndcatng that P gven n (1) s a feasble soluton. It follows that the maxmum utlty of the secondary user s U S = Y log (Y/[C ]), [1,K], (13) where Y = w S [α(d II d III ) αβd II + d III ]. Snce US s a monotone ncreasng functon of Y, and d II d III, t can be verfed that US s a monotone decreasng functon of α and β. Snce α(1 β) and (1 α), US s a monotone ncreasng functon of d II and d III. From a secondary user s perspectve, the best scenaro s α =, β =, and K =3,.e., the entre tme slot s Phase III wth the mnmum number of followers. The selected secondary users enjoy the hghest data rate. The prmary user can only collect revenue from the three secondary users. B. Prmary User Utlty Maxmzaton Gven the optmal strateges of all the secondary users, we substtute f P (R P ) and P nto (9). It follows that U P (α, β, K) =w P ln[αβrp I + α(1 β)rp II ]+ Kw S [α(1 β)d II +(1 α)d III ]. (14) The prmary user solves the followng problem to maxmze ts utlty. max U P (α, β, K, P ). (15) α 1, β 1,3 K K T Maxmzaton of the prmary user utlty s more complcated. We examne the problem for dfferent parameter ranges and derve the local maxmzer n each range. The global optmum s found by comparng the local maxmzers. Wthout loss of generalty, we assume w P = w S. The analyss can be easly extended to the case when w P w S. 1) Case I When K T (N 1): Due to lack of space, we gve the followng Lemma wthout provng t. Lemma 1. For K T N 1, U P acheves ts maxmum when α =1, β =, K =N, and the maxmum value s gven by U 1 P (1,, N ) = w P ln(log(snr)) + w S (N ). (16) ) Case II When K T = (N ): It can be readly concluded that the concluson gven n Secton IV-B1 stll holds. So we have the followng theorem. Theorem 1. When K T N, U P s maxmzed when α =1, β =, K =N, and the maxmum of U P s gven n (16). Note that when K =N, d II =1, d III =1. Theorem 1 ndcates that, when there are plenty of secondary users, to maxmze the prmary user s utlty, we should select N out of them so that each of the selected secondary user can have exactly one nterference free channel. Snce α =1and β =, there s no Phase I and Phase III. To maxmze the prmary user utlty, there s no need for the prmary user to use MIMO transmsson alone. Transmttng data wth dstrbuted nterference algnment whle collectng revenue from spectrum leasng s the best strategy for the prmary user. 3) Case III When 3 K T (N 3): In ths secton, we consder the case when 3 K T N 3. Due to lack of space, we omt the analyss n ths case, whch s smlar to the cases when K T (N 1) and when K T =(N ). We summarze the results n Case III n the followng theorem. Theorem. When K T N 3, U P s maxmzed when (KT +1)(KT +) α = NK T, β =, K = K T, and the maxmum of U P s gven by (17). ( ) UP (KT + 1)(K T +), (17) NK [ T = w P ln log(snr) K ] ( T +1 N + K T w S K T K T +1 1 ). K T C. The Unque Stackelberg Equlbrum We now summarze the analyss n Sectons IV-A and IV-B. The unque Stackelberg Equlbrum of the game defned n Secton III s gven n the followng theorem. Theorem 3. The unque Stackelberg Equlbrum s gven by: (α,β,k )= { (1, (, N ), ) f KT N (KT +1)(K T +) NK T,,K T, f 3 K T N 3 (18) P =[w S α (1 β )d II + w S (1 α )d III ]/C,. (19) Snce we can rewrte (19) as P = w S [α(d II d III ) αβd II + d III ]/C and d II d III, P s a monotone decreasng functon of α and β. On the other hand, P s a monotone ncreasng functon of d II and d III, ndcatng that P s a monotone decreasng functon of K. The secondary users wll adjust ther transmtter power n lght of α, β and K. The best scenaro for them s α =, β =and K =3, for whch there s only Phase III wth the fewest players. Knowng the optmal strateges of the secondary users, the prmary user wll set α =1, β =, and K =N when there are a suffcent number of secondary users. Each selected secondary user has exactly one nterference free channel, and there s only Phase II n the tme slot. In ths case, the prmary

5 user can collect as much revenue as possble whle keepng a relatvely low-rate data transmsson. The secondary users clam s satsfed n part. If there are not as many secondary users as needed, the prmary user wll set the parameters carefully accordng to (18). Under ths condton, the prmary user selects all the secondary users, dscards Phase I, and makes a trade-off between Phase II and Phase III accordng to how many secondary users are there n the system. U P Proposed Scheme when K >=N T Scheme w/o Spectrum Leasng V. SIMULATION STUDY Smulatons are conducted to valdate the performance of the proposed scheme. We frst compare the proposed scheme wth a scheme wthout spectrum leasng to demonstrate the benefts of spectrum leasng. We then compare the proposed scheme wth the cooperatve scheme presented n [9] to demonstrate the effcacy of dstrbuted nterference algnment SNR (db) N Fg.. Utlty of the prmary user n Log scale. A. Wth or Wthout Spectrum Leasng We frst consder the case when there s a suffcent number of secondary users,.e., K T N, snce n many realworld applcatons there are usually more secondary users than the number of antennas at each node. In Fg., we plot the prmary user utlty UP versus the number of antennas N and SNR. In the smulaton, the weght factors are w P = w S =.8. The nose spectral densty s =.1. Note that the maxmum utlty of the prmary user wthout spectrum leasng can be shown to be: UP 3 = w P ln(n log(snr)). () It can be seen from Fg. that there s a huge gap between the proposed scheme and the scheme wthout spectrum leasng. Note that the utlty ncrease due to SNR s less obvous than that due to N, snce the mpact of SNR s dmnshed by the logarthms functons n (3) and (4). Ths clearly ndcates that under the same settng, leasng spectrum to secondary users can greatly mprove the prmary user utlty. Also t can be shown that, the utlty of the proposed scheme s strctly larger than that of no spectrum leasng, for any feasble values of w P, N and SNR. In Fg. 3, we examne the mpact of weght w P on the prmary user utlty UP. We plot the results wth or wthout spectrum leasng, and for N =, 4, and 6. It can be seen that when w P s ncreased, the gap between the proposed scheme and the scheme wthout spectrum leasng becomes larger. Although wth ncreased w P, the prmary user emphaszes more on data transmsson, the revenue s stll ncreased at a hgher speed wth spectrum leasng. The gap also becomes larger when the number of antennas for each node s ncreased. Ths s also because the revenue ncreases faster wth spectrum leasng than the no leasng scheme as N s ncreased. We then consder the case of an nsuffcent number of secondary users. In the smulaton, there are K T =3secondary users. The number of antennas s N =. We plot the prmary user utlty for the proposed scheme and the no-spectrumleasng scheme. We fnd that there s also a bg gan acheved by the proposed scheme. Ths s consstent wth our prevous U P U P Proposed Scheme, N= Scheme w/o spectrum leasng, N= Proposed Scheme, N=4 Scheme w/o spectrum leasng, N=4 Proposed Scheme, N=6 Scheme w/o spectrum leasng, N=6 4 w P Fg. 3. Utlty of the prmary user when K T N. Proposed Scheme when K T <=N 3 Scheme w/o Spectrum Leasng SNR(dB) Fg. 4. Utlty of the prmary user when K T N 3. dscussons. In ths case, the prmary user should stll lease ts spectrum to secondary users to maxmze t own utlty.

6 U P Fg. 5. Scheme w/o Spectrum Leasng Lower Bound of the Proposed Scheme Upper Bound of the Cooperatve Scheme K Comparson of the proposed scheme wth the cooperatve scheme. B. Wth or Wthout Dstrbuted Interference Algnment Fnally, we compare the proposed scheme wth the cooperatve scheme presented n [9]. For a far comparson, we replace the satsfacton functon f P (R P )=1/[1+e a(rp R) ] n [9] wth f P (R P )=ln(r P ). We frst derve an upper bound for the prmary user utlty n [9] (denoted as UP 7 ) usng our notaton. Then we compare the proposed scheme wth the derved upper bound. It follows that [9] where U 7 P = w P ln(r P )+w S (1 α)(k 1)/ (1/R S ), (1) where R P = mn{αβr PS,α(1 β)r SP } R PS = log(1 + mn hps, P ) R SP = log(1 + hp P + R S = log(1 + hs P ), and the h s are channel states. Snce mn h SP, P ) R P = mn{αβr PS,α(1 β)r SP } α R PSR SP <αr PS R PS + R SP α log (1 + P/ ), and R S = log(1 + h S P/ ) log (1 + P/ ), we have: () UP 7 < w P ln[α log(1 + SNR)] + w S (1 α)(1 1/K) log(1 + SNR). (3) Defne f 6 (α) =ln[αlog(1+snr)]+(1 α)(1 1/K) log(1+ SNR). ForSNR 3 (true for the hgh SNR regon), f 6 (α) s maxmzed at ˆα = K/[(K 1) log(1 + SNR)]. Substtute ˆα nto (3), we have: U 7 P <w P ln ( K K 1 ) [ K 1 +w S K log(1+snr) 1 ], (4) whch provdes an upper bound for the the utlty of the cooperatve scheme. In Fg. 5, we plot the smulaton results for the proposed scheme, the cooperatve scheme, and the no-spectrum-leasng scheme. In the smulatons, snce the number of antennas must satsfy N K+ 1, as the number of K vares, we set N = K+. So we are actually comparng the lower bound of our proposed scheme wth the upper bound of the cooperatve scheme. It can be seen from Fg. 5 that both spectrum leasng schemes outperform the no-spectrum-leasng scheme. Furthermore, the proposed scheme outperforms the cooperatve scheme wth consderable gans. Such gans justfy the effcacy of dstrbuted nterference algnment, whch greatly enhance the overall system capacty. VI. RELATED WORK Ths paper s closely related to the research on CR networks. For a general survey of CRs, nterested readers are referred to []. In a CR network, the prmary user s ether aware or unaware of the exstence of the secondary users. Ths paper falls nto the frst category. The prmary user s not only aware of the exstence of the secondary user, but also knows the mpact of the rules on the secondary user behavor. Most of the prevous work, such as [9], [13] [16], only consdered the sngle antenna case, whle we consder multple antennas and explot multplexng gan n ths paper. Ths paper s also related to the research on nterference algnment. In [4], the authors ntroduced the nterference algnment technque. The sgnfcance of ther work s that, by adoptng nterference algnment, the system s no longer nterference lmted. Wth symbol extenson, the system could acheve a normalzed DoF of K/. Another mportant ssue, the feasblty condton, was nvestgated n [11] for structureless generc wreless channels. For wreless channels wth a structure, such as dagonal channels, our recent paper [17] nvestgated the applcaton of nterference algnment n mult-user OFDM networks. To address the concern on the global CSI requrement, a dstrbuted nterference algnment algorthm was proposed n [7], whch only requres local CSI. In [18], nterference algnment and cancellaton were ntegrated to acheve enhance the throughput of MIMO W- F networks. In [19], L et al. proposed a general algorthm for the mult-hop mesh networks. Ths work was motved by these nterestng papers. However, many of the related work manly focused on physcal layer ssues. Ths paper consders how to adopt dstrbuted nterference algnment n a MIMO CR network wth a novel Stackelberg game based approach. VII. CONCLUSIONS In ths paper, proposed a three-phase cooperatve spectrum leasng scheme wth dstrbuted nterference algnment. The system was modeled as a Stackelberg game. Wth backward nducton, we derved the unque Stackelberg equlbrum. Through rgorous analyss, we found the best strateges for the prmary user and secondary users under a broad range of condtons and parameters, and dscussed practcal mplcatons. We also found that leasng spectrum to secondary users s always helpful for enhancng the prmary user utlty.

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