IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 2, FEBRUARY

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO., FEBRUARY The Sum-Capacty of the Ergodc Fadng Gaussan Cogntve Interference Channel Dana Maamar, Natasha Devroye, and Danela Tunnett Abstract Ths paper characterzes the sum-capacty of the ergodc fadng Gaussan overlay cogntve nterference channel EGCIFC, a tme-varyng channel wth two source destnaton pars n whch a prmary/lcensed transmtter and a secondary/ cogntve transmtter share the same spectrum and where the cogntve transmtter has noncausal knowledge of the prmary user s message. The throughput/sum-capacty s characterzed under the assumpton of perfect knowledge of the nstantaneous fadng states at all termnals, whch are assumed to form an ergodc process. A gene-aded outer bound on the sum-capacty s developed and then matched wth an achevable scheme, thereby completely characterzng the sum-capacty of the EGCIFC. The power allocaton polcy that maxmzes the sum-capacty s derved. It s shown that the sum-capacty achevng scheme for an EGCIFC s separable n all regmes.e., codng across fadng states s not necessary, as opposed to the classcal nterference channel. Extensons to the whole capacty regon are dscussed. As a capacty achevng scheme for the EGCIFC under certan channel gan condtons, and as a topc of ndependent nterest, the ergodc capacty of a pont-to-pont multple-nput sngle-output channel wth per-antenna power constrants and wth perfect channel state nformaton at all termnals s also derved. Index Terms Cogntve rado, ergodc capacty, power allocaton, separablty, convex optmzaton. I. INTRODUCTION THERE are two mportant aspects that make wreless communcaton challengng. The frst s fadng, that s, the tme varaton of the channel gans due to small scale effects, such as those due to mult-path, and large-scale effects, such as those due to path loss and shadowng. The second s nterference between wreless users communcatng over the same frequency band. In ths work we focus on the two-user fadng nterference channel where one of the transmtters s cogntve, or knows the message of the other ndependent transmtter. Ths channel model thus experences both fadng and nterference, as well as a thrd phenomena seen n wreless communcatons: the Manuscrpt receved March 4, 014; revsed June 19, 014 and September 3, 014; accepted September 1, 014. Date of publcaton September 5, 014; date of current verson February 6, 015. Ths work was supported n part by the Natonal Scence Foundaton under Award The contents of ths paper are solely the responsblty of the authors and do not necessarly represent the offcal vews of the NSF. The results n ths paper were presented n part at the IEEE Internatonal Symposum on Informaton Theory, Honolulu, HI, USA, June 014. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was L. K. Rasmussen. The authors are wth the Department of Electrcal and Computer Engneerng, Unversty of Illnos at Chcago, Chcago, IL USA e-mal: dmaama@uc.edu; devroye@uc.edu; danelat@uc.edu. Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer /TWC ablty of transmt nodes to cooperate. We am to characterze the sum-capacty/throughput of ths channel so as to hghlght the mpact of fadng, nterference and asymmetrc cooperaton between transmtters. Cogntve networks are wreless networks n whch certan nodes are cogntve rados, or artfcally ntellgent devces, and have been the subject of ntensve nvestgaton by the wreless communcaton communty n the past decade. In a cogntve network, cogntve/secondary transmtters share the spectrum wth prmary/lcensed users. The cogntve devces explot sde nformaton about ther envronment to mprove spectral management. Dependng on the nature of the sde nformaton [], cogntve users ether search for unused spectrum nterweave, or operate smultaneously wth non-cogntve transmtters as long as the nterference produced s wthn an acceptable threshold underlay, or relay part of the prmary user s message and cancel nterference through advanced encodng schemes overlay. In ths paper we consder the overlay paradgm where a prmary transmtter recever Tx Rx par share the same spectrum wth a cogntve Tx Rx par. Accordng to the overlay paradgm [], the secondary Tx s assumed to have non-causal knowledge of the message of the prmary Tx. The channel between the Txs and the Rxs s assumed to experence ergodc fadng.e., tme average of every suffcently long fadng realzaton equals the statstcal average. All nodes n the network are assumed to have perfect nstantaneous knowledge of the channel fadng coeffcents, or full channel state nformaton CSI. Although the full CSI assumpton mght be mpractcal even n the presence of a dedcated feedback channel from the Rxs to the Txs, the resultng model serves as the customary frst step towards understandng the performance of more realstc models. Under these assumptons we seek to determne the sum-capacty of the network and the correspondng optmal power allocaton polcy under a longterm average transmt power constrant at the Txs. In dong so, we also seek to answer the queston of whether codng separately across fadng state s optmal [3]. Ths channel model, whch we term the ergodc fadng Gaussan overlay cogntve nterference channel EGCIFC, s practcally motvated as follows: Channels wth Retransmsson. Consder a network of prmary and cogntve users n whch a fxed-rate,.e., not a functon of the channel gans, prmary user s message was sent but was not decoded at the ntended recever. The prmary recever nforms the prmary transmtter by sendng a NACK, and a retransmsson takes place. If the cogntve transmtter overheard the prmary s ntal IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 810 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO., FEBRUARY 015 transmsson and was able to successfully decode the message n the frst round then, n the retransmsson phase, the cogntve transmtter would have non-causal prmary message knowledge and could transmt together wth the prmary n the ARQ rounds [4]. Coordnated Multpont Transmsson. Consder a network n whch multple transmtters may be thought of as basestatons are connected by hgh capacty backhaul lnks, allowng them to exchange messages to be transmtted to the recevers may be thought of as moble users. Ths model ncludes the EGCIFC as a specal case. In general, studyng networks where nodes may exchange messages through hgh-capacty backhaul lnks may reveal what type of message knowledge structure maxmze the network performance [5], [6]. Overlay Networks. Ths model s nspred by the dea of layered cogntve networks where the frst layer conssts of prmary users and each addtonal layer conssts of cogntve users that share the same spectrum. Each addtonal layer s gven the codebooks of all prevous layers. Ths herarchcal codebook knowledge enables them to causally learn the lower layers messages and ad n ther transmsson. Thus studyng the non-causal message knowledge settng provdes an upper bound to the more realstc case where the messages are causally learned by the cogntve transmtters [7]. A. Pror Work In [8] the two-user Gaussan nterference channel wth ergodc fadng was ntroduced and the optmal power allocaton polcy that maxmzes the outer bound was nvestgated. In [3] the authors also consdered the same model and showed that n general jont encodng and decodng across fadng states s necessary to acheve capacty when perfect CSI at all nodes s assumed note that fadng s an example of the more general result that parallel nterference channels are not separable [9]. However, [3] showed that n some parameter regmes, such as very strong or very weak nterference, separate encodng and decodng across fadng states s optmal. In other words, n nterference channels separablty may hold for certan channel states but not for all channel states. Whle nterference channels are not separable n general, t s known that mult-access [10] and broadcast [11] channels wth the same CSI assumpton are separable. Snce the EGCIFC has elements of both an nterference channel and a broadcast channel, t s not apror obvous that a separable scheme s capacty achevng. A formal proof of the optmalty of separablty for the EGCIFC s gven n ths paper. In [1] the authors also consder a fadng cogntve network under the underlay paradgm. The prmary user n ths case s completely oblvous to the exstence of the cogntve user. The relatonshp between the achevable capacty of the secondary channel and the nterference caused at the prmary recever was quantfed. Instantaneous and average nterference power constrants were both consdered and the optmal power allocaton polcy for the secondary user n each case was derved. In ths work we consder the overlay paradgm wth long-term average power constrants at the prmary and secondary transmtters. Moreover our prmary user s not oblvous to the exstence of the secondary user. In [13] the authors consder a cogntve rado network under the underlay paradgm where prmary and secondary users are subject to block fadng. The prmary user s not capable of adaptng ts power allocaton whle the secondary user s able to do so. The authors derve the optmal power allocaton strateges for the cogntve user to maxmze ts ergodc and outage capacty. In our work both users are capable of adaptng ther transmt power over the dfferent fadng states based on the channel state nformaton, and we consder the overlay paradgm. In [14] the authors consder the fadng cogntve sngle nput sngle output SISO MAC channel for the underlay paradgm, where the secondary users are subject to both a transmt power constrant and nterference power constrant to prmary users. It s shown that the sum-capacty achevng power allocaton polcy s a water fllng type of soluton. The key dfference between the EGCIFC and the model n [14] s the message structure; n [14] each sender only knows ts own message and thus the transmt sgnals are ndependent; n the EGCIFC the transmt sgnals are correlated due to the non-causal prmary message knowledge at the secondary transmtter. Because of ths, the sum-capacty achevng power allocaton polcy for the EGCIFC wll not be a water fllng type. As a byproduct of our analyss and as a result of ndependent nterest, we shall show that the optmal power polcy for the EGCIFC n some regmes requres both transmtters to beamform to the prmary recever; n ths case the model reduces to a pont-to-pont multple nput sngle output MISO channel wth per-antenna power constrants PerPC. In [15] the author fnds the capacty for the pont-to-pont MISO channel wth PerPC wth two dfferent assumptons of CSI. The capacty for a constant channel wth CSI at both the transmtter and recever and that of a Raylegh fadng channel wth CSI at the recever only were derved. The optmal sgnalng scheme was found for both cases. The author compares the result wth the capacty of a MISO channel wth sum power constrants SumPC through numercal examples. In both cases the capacty wth PerPC s, as expected, less than that wth SumPC. Here we characterze the sum-capacty of the EGCIFC wth CSI at the transmtters and recevers and fnd the optmal sgnalng scheme, whch s thus dfferent from the solutons found n [15]. In [16] the authors frst ntroduced the nformaton theoretc study of the cogntve rado channel same as the cogntve nterference channel whch falls nto the overlay paradgm. In that work, the channel gans were constant and achevable rate regons and outer bounds were derved. In [17] the authors found the capacty of the cogntve nterference channel n the weak nterference regme. In partcular, the power splt whch ensures that the prmary recever rate contnues to be the same as that wthout nterference from the cogntve user. For the state-of-the-art on the two-user cogntve nterference channel wth constant channel gans known to all nodes we refer the reader to [18] and [19]. In ths paper we remove the assumpton of constant channel gans and consder the fadng tme varyng cogntve nterference channel.

3 MAAMARI et al.: SUM-CAPACITY OF THE EGCIFC 811 B. Contrbutons and Paper Outlne Our man contrbutons and paper organzaton are as follows. Secton II presents the channel model for the EGCIFC. Secton III contans the followng man results: 1 We frst present a sum-capacty gene-aded outer bound for the EGCIFC. Ths gene-aded outer bound conssts of gvng sde-nformaton to the cogntve recever of prmary user s message and output to the cogntve recever, as for the non-fadng case [18], [19]. We then provde a matchng achevablty scheme as an extenson to the fadng case of the scheme n [17], thus completely characterzng the sum-capacty under the assumpton of perfect CSI at all nodes. When the prmary recever experences strong nterference, the sum-capacty achevng scheme s that of pontto-pont MISO channel wth PerPC and perfect CSI at the transmtter and recever. In [1] we have characterzed the capacty of a MISO wth full CSI at all nodes and wth arbtrary number of antennas. In ths work, we further consder the case of weak nterference whch was left open n [1]. 3 The power allocaton polcy that maxmzes the sumcapacty for the EGCIFC s derved and shown to depend on the relatve channel gans for a gven fadng state. Extensons to the whole capacty regon are dscussed. In Secton IV we llustrate our results wth numercal examples. We consder a Raylegh fadng pont-to-pont MISO channel wth PerPC. The capacty for ths channel s compared to that of a MISO channel wth SumPC and that of a MISO channel wth constant power allocaton. We then consder a Raylegh EGCIFC where the mean of the channel gans are chosen such that wth hgh probablty the channel s ether n strong nterference or n weak nterference. The sum-capacty of the EGCIFC under the optmal power allocaton s compared to that wth constant power allocaton and that of a MISO channel wth PerPC.e., allocate power as f the channel s n strong nterference even f ths s not the case. We then consder a scheme where we optmze the correlaton coeffcent or power splt among the nputs but keep the power allocaton constant at each fadng state and show that, although n prncple ths s a suboptmal scheme, t s almost capacty achevng for the EGCIFC thereby hghlghtng the mportance of correctly balancng the amount of power the secondary user uses for transmsson of ts own message and that for relayng the prmary message. Some proofs may be found n Appendx. C. Notaton Throughout the paper we use the followng notaton: x n denotes a vector of length n wth components x 1,...,x n. A represents the transpose and complex-conjugate of the matrx A. A represents the optmal soluton for a gven optmzaton problem. A N represents a vector of random varables A wth [1 : N]. Fg. 1. Ergodc fadng Gaussan cogntve nterference channel EGCIFC. E[.] denotes the expectaton operator. log. denotes logarthm n base, [x] + := maxx, 0 and log +. := maxlog., 0. I.;. denotes the mutual nformaton and h. denotes the dfferental entropy. PA represents the probablty of event A. N μ, σ denotes a complex-valued crcular symmetrc Gaussan random varable wth mean μ and varance σ. II. CHANNEL MODEL The cogntve nterference channel conssts of two transmtreceve pars Tx 1 to Rx 1 and Tx to Rx representng the cogntve and prmary users, respectvely, as shown n Fg. 1. Each transmtter Tx k wshes to convey to ts destnaton Rx k an ndependent message W k, whch s unformly dstrbuted over the set [1 : NR k ], where R k s the rate n bts per channel use, and N represents the codeword length, for k [1 : ]. Tx 1 s cogntve n the sense that t has non-causal message knowledge of the prmary user s Tx message W.Arate vector R 1,R s sad to be achevable f there exsts a famly of codes ndexed by N such that the probablty of decodng error can be made arbtrarly small [0]. The sum-capacty s defned as the maxmum achevable R 1 + R. In Gaussan nose and wth ergodc fadng, the ECIFC nputoutput relatonshp at every tme nstant tme ndex s omtted for easer notaton s gven by Y 1 = h 11 X 1 + h 1 X + Z 1, Z 1 N0, 1, 1 Y = h X + h 1 X 1 + Z, Z N0, 1, [ ] h11 h where H := 1 denotes the random channel gan h 1 h matrx wth complex entres generated randomly at each tme nstant/channel use accordng to a known, statonary and ergodc random process, wth [H],j = h j C,,j [1 : ] representng the fadng channel gan between Tx j and Rx.A realzaton of H s ndcated as h. Tx j, wth channel nput X j s subject to the long-term average power constrant E[ X j ] P j,j [1 : ]. Wth CSI at all termnals, the transmtters can perform dynamc power allocaton, and transmt wth power P j h 0, j [1 : ], at a channel use wth fadng state h.we seek to determne the sum-capacty optmal power allocaton for each user such that E[P j h] P j, j [1 : ]. Thanks to cognton, the channel nputs can be correlated; the nput covarance matrx n fadng state h s denoted by [ P 1 h ρ h ] P 1 hp h Σh:= ρh, 3 P 1 hp h P h where the correlaton coeffcent must satsfy ρh 1.

4 81 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO., FEBRUARY 015 III. MAIN RESULTS Ths secton ncludes the man results for the EGCIFC. A sum-capacty outer bound s presented as a maxmzaton problem over three constrants: the two long-term average power constrants at the transmtters and the constrant on the correlaton coeffcent. We then prove that the outer bound wth the optmal power allocaton polcy s achevable through a varaton of the achevablty scheme of [17] proposed for the constant channel gan cogntve nterference channel n weak nterference. The achevablty scheme, unlke that for certan parameter regmes of the nterference channel, s separable, that s, encodng need not be done across fadng states. For a certan parameter regme relatve channel gans, the sum-capacty achevng power allocaton corresponds to the optmal power allocaton scheme for a pont-to-pont MISO channel wth a PerPC. Thus as a topc of ndependent nterest we characterze the power allocaton polcy and the capacty of the fadng pontto-pont MISO channel wth PerPC and an arbtrary number of antennas. The sum-capacty of the EGCIFC s presented next and s shown to be the soluton of a maxmzaton problem n the varables P 1 h,p h,ρh n 3. We have: Theorem 1: The ergodc sum-capacty of the EGCIFC s [ C EGCIFC, sum =maxe log 1+h Σhh 1+1 ρh max{ h 11, ] }P 1 h +log 1+1 ρh, 4 P 1 h wth h := [h 1 h ] the ndex refers to the second row of the nstantaneous fadng realzaton channel matrx h and where the maxmzaton s over P h 0:E[P h] P, [1 : ], and ρh 1. Proof: As a generalzaton of the outer bound technque of [18], [19] to the fadng case, the sum-capacty s upperbounded by NR 1 + R ɛ N a I W 1 ; Y1 N h N + I W ; Y N h N b I W 1 ; Y1 N,Y N h N,W + I W ; Y N h N c = I W 1,Y1 N h N,W,Y N + I W1 ; Y N W, h N + I W ; Y N h N d = I W 1 ; Y1 N h N,W,Y N + I W1,W ; Y N h N e I X1 N ; Y1 N h N,X N,Y N + I X N 1,X N ; Y N h N f = h Y1 N h N,X N,Y N h Y N 1 h N,X N,Y N,X1 N + h Y N h N h Y N h N,X1 N,X N g = h Y 1 h N,X N,Y N, Y =1 h Y 1 h N,X N,Y N,X1 N, Y h Y h N,Y 1 1 h Y h N,X1 N,X N, Y 1 h hy 1 h,x,y h Y 1 h,x,y,x 1 =1 + hy h hy h,x 1,X 1 = j IX 1 ; Y 1 X,Y, h +IX 1,X ; Y h =1 IX 1G ; Y 1 Y,X G, h +IX 1G,X G ; Y h, =1 where the dfferent nequaltes follow from: a Fano s nequalty here ɛ N 0 as N, b a gene provdes sde nformaton Y N,W to Rx 1 and ndependence of messages, c chan rule for mutual nformaton, d recombnng mutual nformaton terms, e data processng nequalty and defnton of encodng functons, f defnton of mutual nformaton, g chan rule for entropy, h condtonng reduces entropy and memoryless channel, defnton of mutual nformaton, and j Gaussan maxmzes entropy, where X 1G,X G are jontly Gaussan wth the same covarance matrx as X 1,X.The dependence on the tme ndex can be elmnated by takng the approprate lmt over N as done n [3]. We note that n g we can choose the correlaton coeffcent among Z 1 and Z snce the Rxs do not cooperate and hence the capacty regon only depends on the nose margnal dstrbutons.e., we can choose the worst nose correlaton as long as the margnal dstrbutons are preserved. From the results on the statc{ channel [19] } we know that the worst nose correlaton s mn h11 h 1, h 1 h 11. Wth ths worst nose correlaton and wth the nput covarance as n 3, the sum-capacty outer bound n g for the EGCIFC can be expressed as n 4. We now demonstrate that the derved outer bound s achevable. A varable rate codng scheme s used: n ths case at each channel use each coordnate of the codewords X1 N and X N, the optmal powers for that partcular fadng state are used P1 h and P h by the secondary and prmary nodes, respectvely wth an optmal ρ h. The cogntve transmtter assgns part of ts power to relay W and uses the remanng power to send ts own message by Drty Paper Codng DPC [1] aganst W, whch t knows non-causally. Smlar to the scheme for the statc channel [17] wth the dfference that at each channel use, the optmal parameters P1 h, P h and ρ h for that partcular fadng state that maxmze 4 are used. In partcular, let U 1 and U be ndependent Gaussan random varables wth zero mean and unt varance, and Prmary user sends X = P hu, 5a Cogntve user sends X 1 = X R + X 1DPC where X R = ρ h P1 hej h 1+ h U, 5b X 1DPC = 1 ρ h P 1 hu 1, 5c and where X 1DPC s DPC aganst the non-causally known state S = h 1 P h+h 11 ρ h P1 hej h 1+ h U.

5 MAAMARI et al.: SUM-CAPACITY OF THE EGCIFC 813 The receved sgnals are Y 1 = h 11 1 ρ h P 1 hu 1 + S + Z 1, Y = e j h h P h+ h 1 ρ h P 1 h U + h 1 1 ρ h P 1 hu 1 + Z ; 5d Snce Tx 1 used DPC we have R 1 =log 1+ h 11 1 ρ h P 1 h, 5e and f Rx treats U 1 as nose we have h 1 ρ h P1 h+ h P h R =log ρ h P1 h. 5f By summng 5e and 5f, re-arrangng and takng the expectaton yelds the sum-rate n 4 when h 11. When h 11 < the sum-rate n 4 n maxmzed by ρ h =1and s agan achevable by 5e and 5f. Remark 1: One can thnk of parallel Gaussan cogntve nterference channels PGCIFC as an EGCIFC n whch each sub-channel occurs wth equal probablty and so the sumcapacty result descrbed n ths paper gves also the sumcapacty for PGCIFC. Whle Theorem 1 expresses the ergodc sum-capacty as an optmzaton problem, we now proceed to determne the optmal power allocaton polcy. To do so, we frst nvestgate a topc of ndependent nterest: the ergodc capacty of the pont-to-pont MISO channel wth PerPC, whch gves the sumcapacty of the EGCIFC when h 11 <. A. The Ergodc Capacty of the Pont-to-Pont MISO Channel Wth Per-Antenna Power Constrants and Perfect CSI at All Termnals The MISO channel wth n transmt antennas wth PerPC has output Y =[H 1 H H n ] X + Z C, Z N0, 1, where each entry of the nput vector X := [X 1,,X n ] T has a separate long-term average transmt power constrant E[ X ] P, for [1 : n]. The channel vector [H 1 H H n ] has complex-valued entres representng the channel gan coeffcent from each transmt antenna to the receve antenna and s generated from an ergodc process whose nstantaneous realzaton s known to the transmtter and the recever. We am to characterze the ergodc capacty of ths channel, where capacty s defned as usual [0]. In the followng, we denote the nstantaneous realzaton of the channel vector as h := [h 1 h n ] C n and the power allocated on antenna n fadng realzaton h as P h, [1 : n]. Theorem : The ergodc capacty of the Gaussan fadng MISO channel wth PerPC s C MISOPerPC =max E log 1+ h P h P j h = [1:n] 6 where the maxmzaton n 6 s over P h 0:E[P h] P, [1 : n]. The optmal power allocaton polcy s gven by [ ] h + [1:n] λ 1 [1:n] h λ h j, 7 j where the Lagrange multplers {λ, [1 : n]} solve the nonlnear system of equatons E[Pj h] = P j, j [1 : n], and attans C MISOPerPC = E log + h. 8 λ [1:n] Proof: The proof s based on solvng the dual problem to 6 and s provded n Appendx A. The capacty n 8 can be obtaned by beamformng: each antenna transmts X =exp{ j h } h U, U N0, 1, [1 : n] P where P h s the optmal power allocaton gven by 7. By takng the average over all fadng states, the capacty can be expressed as 8. Remark : If Lagrange multplers n 7 are all equal to λ, then the power allocaton becomes [ 1 P h = λ 1 ] + h h h, 9 wth h =: [1:n] h. The expresson n 9 corresponds to the water-fllng power allocaton optmal under SumPC, n whch case the Lagrange multpler would satsfy [ ] E [1:n] P h = E [ [ 1 λ 1 h ] + ] = [1:n] P j.ths can happen f the power constrant on each antenna s the same and the dstrbuton of the fadng vector does not change by permutng ts components, such as wth dentcal and ndependent dstrbuted fadng. Remark 3: In [15] the capacty of the fadng MISO channel wth PerPC was derved analytcally under the assumpton of CSI at the recever only n Theorem we consder the case of CSI at both the transmtter and recever, and obtan the capacty wth PerPC n closed-form. Determnng analytcally the capacty under PerPC s elusve because the correspondng Lagrangan dual problem does not seem to lead to a closed form soluton n general. For example, n [] the authors consdered a MIMO-MAC wth per-antenna power constrants and could only fnd effcent algorthms to solve numercally the problem of fndng the optmal nput covarance matrces that maxmze the sum-capacty. The capacty of the statc pont-to-pont MIMO channel wth PerPC

6 814 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO., FEBRUARY 015 TABLE I OPTIMAL POWER POLICY WHEN < h 11 was derved n closed form for certan full-column-rank channel matrces n [3], whle n [4] condtons for optmalty of beam-formng under PerPC were derved together wth an algorthm for optmal mult-stream beam-formng. Remark 4: The capacty of the fadng MISO pont-to-pont channel wth PerPC can not be deduced from capacty results for the fadng SISO MAC [5]. Although one could thnk of a user n the fadng SISO MAC as an antenna n the fadng MISO pont-to-pont channel, the analogy stops there; the reason s that n the fadng SISO MAC the users send ndependent nputs whle n the fadng MISO pont-to-pont channel the sgnals sent by the antennas can be correlated. As a result, the sumcapacty achevng power polcy for the fadng SISO MAC s the followng water-fllng soluton P h = [ 1 1 ] + λ h f h λ = max k=1,...,k { hk λ k }. 10 whch does not correspond to 7 for example, under 7 ether all the antennas send wth a strctly postve power or all stay slent, whle under 10 at most one user/antenna sends at any gve tme. B. Fadng Cogntve Interference Channel The sum-capacty of the EGCIFC n 4 nvolves a maxmzaton over the power allocaton polcy of both transmtters and a correlaton coeffcent between the nputs over dfferent fadng states. We now seek to solve ths optmzaton problem, whch n turn depends on the relatve strengths of the channel gans between the transmtters and recevers n the channel. We have the followng theorems that descrbe the optmal soluton. Theorem 3 Strong Interference at the Cogntve Recever/Rx : When h 11, the optmal power allocaton polcy for the EGCIFC n Theorem 1 corresponds to that of pont-to-pont MISO wth PerPC n Theorem. Proof: Gven that h 11 s satsfed, then t s clear that ρ h =1s optmal n 4. We are then left wth solvng for the optmal power allocaton. Settng ρ h =1 reduces the optmzaton problem n 4 to that n 6 and hence the optmal power allocaton strategy s gven by Theorem. Therefore, settng R 1 =0 turns out to be optmal for the EGCIFC n ths regme,.e., the best use of cogntve user s ablty s to broadcast the prmary s message. Theorem 4 Weak Interference at the Cogntve Recever/Rx : When < h 11, the optmal power allocaton polcy for the EGCIFC s summarzed n Table I and s one of ether of the followng polces: 1 both users refran from transmttng, cogntve transmtter water-flls over h 11, 3 prmary transmtter water-flls over h, 4 MISO wth SumPC type of power allocaton, or 5 both users transmt to ther ntended recevers wth non-zero powers n ths case the optmal polcy must be determned numercally. Proof: The proof, based on solvng the Lagrangan dual problem of 4, s provded n Appendx B. One may nterpret the polces n Table I as the followng. For R 1 both drect lnk channel gans are weak smaller than the correspondng optmal Lagrange multpler and so the optmal scheme for both transmtters s to refran from allocatng power, savng the power for better channel states. In R the cogntve transmtter water-flls over ts drect lnk h 11 whle the prmary user refrans from allocatng any power because n ths regme h 11 s not weak whle h s weak. One can nterpret R 3 smlarly to R but wth the roles of the users swapped. h 11 The channel gan condton n R 4 mples that gven that h 1 > 1; n ths case the sum of the channel gans to the prmary recever s stronger than that of the drect gan to the cogntve recever and performng a pont-to-pont MISO-type power allocaton s optmal. In R 5 both transmtters send wth non-zero power; n ths case a closed-form soluton for the optmal power allocaton polcy s not avalable. Remark 5: The above analyss showed that a separable achevable scheme s sum-capacty optmal. In order to characterze the whole capacty regon one needs bounds on R 1 and R too; the cut-set approach gves such bounds. Therefore the capacty regon of the EGCIFC s outer bounded by R 1 ɛ N 1 N R ɛ N 1 N R 1 + R ɛ N 1 N IX 1G ; Y 1 X G, h =1 IY G ; X 1,X G h =1 IY ; X 1G,X G h =1 + IX 1G ; Y 1 X G,Y, h,

7 MAAMARI et al.: SUM-CAPACITY OF THE EGCIFC 815 where the last bound s from Theorem 1 and the sngle rate bounds are cut-set bounds, smlarly to [19, eq. 8]. As for the sum-capacty, the whole regon s exhausted by consderng jontly Gaussan nputs and where the regon can be tghtened by choosng any Y Y. By further takng the approprate lmt over N as done n [3] and by consderng the nput covarance as n 3, the upper bound regon can be expressed as R 1 E [ log 1+ h 11 P 1 h 1 ρh ], 11a [ ] R E log 1+h Σhh 11b [ R 1 + R E log 1+h Σhh ] 1+ 1 ρh max{ h 11, }P 1 h +log 1+1 ρh, P 1 h 11c By consderng the achevable scheme n [19, eqs. and 3], whch was shown to be at most to wthn 1 bt per channel use per user of the capacty regon outer bound for the statc/non-fadng case, the outer bound n 11 can be shown to be achevable to wthn 1 bt per channel use per user as well as n the EGCIFC. We note that ths achevable scheme nvolves two DPC steps, one per user whle the scheme n 5 that only has one DPC step and t s only approxmately optmal to wthn a constant gap whle the scheme n 5 s exactly sum-rate optmal. We therefore conclude that, n order to characterze the whole capacty regon of the EGCIFC to wthn a gap, t suffces to characterze the closure of the outer bound n 11, whch can be done by solvng the followng famly of convex optmzaton problems: for each λ [0, 1] C EGCIFC,regon λ :=max{λr 1 +1 λr }, 1 where the maxmzaton s over the rate pars R 1,R n 11. Solvng the optmzaton problem n 1 s not a trval extenson of the sum-capacty results presented n Theorems 3 and 4 and s beyond the scope of ths paper. Ths s so because one needs to consder whch bounds are actve n 11 n order to determne the optmal corner pont as a functon of λ [0, 1]; the coordnate of such a pont must be plugged n the optmzaton problem n 1 and the correspondng KKT condtons must be worked out smlarly to Appendx B. We expect that there wll be regmes n whch the KKT condtons must be solved numercally as for the sum-capacty. IV. NUMERICAL RESULTS In ths secton we numercally evaluate Theorems 1 and for the case where the channel gans are ndependent Raylegh random varables, not necessarly wth the same mean parameter. A. The Pont-to-Pont MISO Channel Wth PerPC We frst consder a 1 pont-to-pont MISO channel. The channel vector [h 1,h ] at each channel use has ndependent Fg.. The capacty of the pont-to-pont MISO channel wth PerPC surface n green, from 13 s upper bounded by that of a MISO channel wth SumPC surface n red, from 16 and lower bounded by that of a MISO channel wth constant power allocaton and dependent nputs surface n yellow, from 14 and wth constant power allocaton and ndependent nputs surface n blue, from 15. and exponentally dstrbuted components wth means γ 1 = E[ h 1 ]=5and γ = E[ h ]=. The transmt antennas are subject to the average power constrants P 1 and P.InFg. four surfaces representng the capactes of dfferent MISO channels are plotted as a functon of the average transmt antenna power constrants P 1 and P. The surfaces corre spond to: 1 the MISO channel wth PerPC C MISOPerPC [ ] = E log 1+ h 1 P h 1 h+ P h 13 where P1 h and P h are gven n 7 for j [1 : ]; the MISO channel wth constant power allocaton and beam-formng where the nstantaneous phases are known to the transmtters to allow for coherent beamformng [ ] C dep,cte = E log 1+ h 1 P 1 + h P ; 14 3 the MISO channel wth constant power allocaton and ndependent sgnalng nstantaneous phases are not known at the transmtter and they are ndependent and unformly dstrbuted n [0, π] as n [15] C ndep,cte = E [ log 1+ h 1 P 1 + h P ] ; 15 and 4 the MISO channel wth a SumPC C MISOSumPC [ ] = E log 1+ h 1 P h 1 h+ P h 16 where P 1 h and P h are gven n 9 for [1 : ].

8 816 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO., FEBRUARY 015 Fg. 3. The sum-capacty of the EGCIFC whch s skewed to be n strong nterference wth hgh probablty surface n red s plotted wth the sumcapacty acheved when consderng a MISO achevablty wth PerPC surface n blue. The MISO power allocaton s almost sum-capacty achevng as the two surfaces almost overlap. The constant power allocaton scheme wth dependent nputs surface n yellow s agan a lower bound on the sum-capacty of EGCIFC. Note that perfect CSI at both transmtters s needed for coherent beam-formng. Numercal evaluatons show that the capacty of the pont-topont MISO wth PerPC s upper and lower bounded by that of the MISO wth SumPC and that of constant power allocaton respectvely. Also as expected, dependent constant nputs n 14 outperform ndependent constant nputs n 15. Remark 6: In [15] the pont-to-pont MISO channel wth PerPC wth Raylegh fadng wth no CSI at the transmtters was compared to that wth SumPC and wth ndependent sgnalng; the author noted that C MISOPerPC = C ndep,cte because of the phase beng ndependent and unformly dstrbuted n [0, π]. Snce here we assume the transmtter has CSI we have C MISOPerPC C ndep,cte. Remark 7: In the case of dependent nputs and beamformng, the channel s assumed to have CSI at the transmtter to account for the channel gan phases and the ablty to coherently beam-form. Ths explans why C MISOPerPC s almost the same as C dep,cte, whch may have practcal mplcatons. B. The EGCIFC Sum-Capacty We now consder two dfferent scenaros for the EGCIFC correspondng agan to Raylegh fadng channels wth dfferent means: Case 1 the means are chosen such that the channel experences strong nterference wth hgh probablty, and Case the means are chosen to experence weak nterference wth hgh probablty. Monte Carlo smulatons were used to evaluate the capactes. In Fg. 3 the sum-capacty for the EGCIFC havng h 11, and h exponentally dstrbuted wth γ 0 = E[ h 11 ]=1, γ 1 = E[ ]=5 and γ = E[ h ]= skewed wth hgh probablty to be n strong nterference snce P[ h 11 ]= γ 1 γ 1 +γ 0 = 5 6 s plotted along wth the sum-capacty of the system usng the MISO wth PerPC transmt strategy. The latter s not optmal n general when the channel experences weak nterference states. Consderatons Fg. 4. The sum-capacty of EGCIFC whch s skewed to be n weak nterference wth hgh probablty surface n green s plotted wth the sum-capacty acheved when consderng a MISO achevablty scheme wth PerPC. The power allocaton as n the MISO wth PerPC surface n green s not optmal as n the case when the channel s skewed to be n strong nterference. smlar to those made n Remark 6 may be made here; moreover the surface representng the sum-capacty C MISOPerPC approaches that of C EGCIFC as expected snce we are skewed to be n strong nterference where the scheme correspondng to a MISO channel wth PerPC s optmal. In Fg. 4 the sum-capacty for the EGCIFC havng mean parameters γ 0 =5, γ 1 =1 and γ = s plotted skewed wth hgh probablty to be n weak nterference P[ < h 11 ]= 5 6 along wth the sum-capacty acheved by usng the power allocaton correspondng to that of a MISO channel wth PerPC. As expected, the MISO scheme wth PerPC not optmal for weak nterference does not perform as well as n the regme where the channel s skewed to be n strong nterference wth hgh probablty. In Fg. 5 we choose the channel gans to be dentcally dstrbuted wth mean parameter γ 0 = γ 1 = γ =1and plot the sum-capacty of the EGCIFC and an achevablty scheme correspondng to constant power allocaton, but wth the optmal correlaton coeffcent whch changes wth each fadng state,.e., the soluton of [ C sum = E max ρh 1 log 1+ P 1 + h P + ρh P 1 h P 1+1 ρh max{ h 11, ] }P 1 +log. 1+1 ρh P 1 Ths s solved for the optmal correlaton coeffcent numercally. It s nterestng to note that by optmzng the correlaton coeffcent only and not the power allocaton,.e., keepng the power constant one can approach a dfference of around 0.3 bts/channel use, at least for these channel condtons, the sum-capacty of the EGCIFC. Ths may have mplcatons n practce.e., constant power may be good enough f one optmally fnds the correlaton coeffcent. We note however that to obtan the optmal correlaton coeffcent at each fadng state, full CSI s stll requred at both transmtters.

9 MAAMARI et al.: SUM-CAPACITY OF THE EGCIFC 817 By takng partal dervatves of L we obtan for every [1 : n] L P h = θ h 1+θ λ =0, P θ := h P h, P h = θ h 1+θ, λ 17 Fg. 5. Two surfaces representng the sum-capacty of EGCIFC wth the optmal power allocaton red and the sum-capacty whle consderng constant power allocaton and an optmzed correlaton coeffcent at each fadng state blue. Although the optmal power allocaton was not utlzed, the blue surface s almost capacty achevng. V. C ONCLUSION In ths work we characterzed the sum-capacty of the ergodc fadng Gaussan cogntve nterference channel. A separable scheme power allocaton depends only on the current fadng state and codng need not be done across fadng states was shown to be optmal. Ths s n contrast to the classcal nterference channel, whch s not separable, but smlar to the fadng broadcast channel and a one-sded fadng nterference channel. The optmal power polcy n strong nterference was shown to be that correspondng to a MISO pont-to-pont channel wth per-antenna power constrants. As a sde result of ndependent nterest, we derved n closed-form the ergodc capacty of ths channel model. Numercal results show that, at least for certan channel parameters, optmzng the correlaton coeffcent between the nputs of the two users, whle keepng the powers fxed, performs nearly as well as optmzng both the powers and correlaton coeffcent. We dscussed how the present work can be extended to the characterzaton of the whole regon, whch s the subject of current nvestgaton. Extensons to settngs wth an arbtrary number of users s also an nterestng drecton for future work. whch mples θ = h P h = θ 1+θ 1+θ = θ = h λ 18 h λ h 1. 0 λ From 17, the optmal power for antenna [1 : n] becomes P j h = + h λ 1 1 λ j P j h = 1 and thus the capacty s [ [ E log 1+ h λ h j λ j 1 h λ + hj λ j ] + ] [ h 1 = E log + λ h λ, h λ ] h, λ and the Lagrange multplers from solve the non-lnear system of equatons [[ ] + ] 1 1 h j λ j P j = E 1. 3 h λ j λ APPENDIX A. Proof of Theorem The pont-to-pont MISO capacty wth PerPC and n transmt antennas s the soluton of the followng optmzaton problem: E log 1+ h P h. max P h 0:E[P h] P, [1:n] The Lagrange dual problem, for λ 0, [1 : n], s L=E log 1+ h P h λ P h P. B. Proof of Theorem 4 The sum-capacty of the EGCIFC s gven n 4. When h 11, the sum-capacty n 4 s clearly maxmzed by ρ h =1and thus we must solve max P 0:E[P h] P, [1:] E log 1+ h P h. 4 The problem n 4 s that of fndng the optmal power allocaton for a pont-to-pont MISO wth PerPC and the soluton was presented n Appendx A. When h 11 >, 5

10 818 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO., FEBRUARY 015 the Lagrangan dual of the sum-capacty n 4 s L = E [ log 1+h Σhh From 6j : h1 P 1 h h P h 1+ P 1 h+ h P h = γ 1h 0. 7c 1+ 1 ρh h 11 P 1 h +log 1+1 ρh P1 h P 1 h P 1 P h P + μ1 hp 1 h+μ hp h ] + γ 1 h ρh γ h1 ρh. whose KKT condtons are μ hp h =0, [1, ], 6a γ 1 h ρh =0, 6b γ h1 ρh =0, 6c P h 0, [1, ], 6d ρh 1, 6e ρh 0, 6f γ h,μ h,λ 0, [1, ], 6g L P 1 h = h 1 + ρh h1 h P h P 1 h 1+h Σhh h11 1 ρh ρh P 1 h 1+ 1 ρh h 11 P 1 h μ 1 h = 0, 6h L P h = h + ρh h1 h P1 h P h 1+h Σhh μ h = 0, 6 L ρh = h1 P 1 h h P h 1+h Σhh h11 ρh P 1 h 1+ 1 ρh P 1 h 1+ 1 ρh h 11 P 1 h +γ 1 h γ h = 0. 6j We next analyze dfferent possble solutons. 1 On the Optmalty of ρh =0.e., Independent Inputs: If ρ h =0 s an optmal soluton then γ 1 h 0 and γ h =0from 6b, 6c and 6g and the followng KKT condtons must hold From 6h : + 1+ P 1 h+ h P h h P 1 h 1 + h 11 P 1 h = μ 1 h, From 6 : 7a h 1+ P 1 h+ h P h = μ h, 7b From 7c we have that P 1 h P h =0, that s, the powers cannot be smultaneously strctly postve. We now proceed by fndng the optmal power allocaton. Subcase B1.1 P 1 h =0and P h =0: From 6a : P 1 h =0 μ 1 h 0 and P h =0 μ h 0, 8a From 7b : h = μ h h 1, 8b From 7a : h 11 = μ 1 h h c We therefore conclude that P1 h,p h, ρ h = 0, 0, 0 s optmal when R 1 := {eq.5 holds, h 11 1, h } 1. Subcase B1. P 1 h > 0 and P h =0: From 6a : P 1 h > 0 μ 1 h =0and P h > 0 μ h 0, 9a From 7a : 1+ P 1 h + h P 1 h 1 + h 11 P 1 h = P 1 h = 1 1 h 11 =: P 1ah, 9b h From 7b : 1+ P 1 h = μ h h P 1 h = μ h 1 =: P 1bh. 9c By mposng P 1a h =P 1b h and snce μ h 0, we obtan 1+ h 1 1 h 11 + h h 1. We therefore conclude that [ ] + P1 h,p h, ρ 1 h = 1 h 11, 0, 0 s optmal when R := eq.5 holds, h 11 >1, h 1+ h 1 h 1 Subcase B1.3 P 1 h =0and P h > 0: 1. h 11 From 6a : P 1 h =0 μ 1 h 0 and P h > 0 μ h =0, From 7a : 1+ h P h + h 11 = μ 1 h, 30a h From 7b : 1+ h P h = P h = 1 1 > 0. 30b h

11 MAAMARI et al.: SUM-CAPACITY OF THE EGCIFC 819 By evaluatng 30a for P h n 30b gves 1+ h + h 11. We therefore conclude that P1 h, [ ] + P h, ρ 1 h = 0, 1 h, 0 s optmal when R 3 := {eq.5 holds, h > 1, h + h 11 h 1 } 1} On the Optmalty of ρ h =1.e., Identcal Inputs up to Affne Transformaton: If ρh =1s an optmal soluton then γ 1 h =0and γ h 0 from 6b, 6c and 6g and the followng KKT condtons must hold. From 6h: + h P h P 1 h = μ 1 h, 31a 1+h Σhh From 6 : From 6j : h + h P1 h P h = μ h, 1+h Σhh h1 P 1 h h P h 1+h Σhh h 11 P 1 h.. 31b 31c If P 1 h P h =0 n 4 then trvally ρ h =0 s optmal as n the prevous case and thus under the assumpton that ρ h =1 we only need to consder P 1 h P h > 0. If both powers are strctly postve we further have that μ 1 h =μ h =0and thus the system of equatons n 31 s equvalent to h1 ξ := P 1 h+ h P h 1+ P 1 h+ h P h, From 31a : h 1 ξ = P1 h, 3a From 31b : h ξ = P h, 3b h1 From 31c : P 1 h h P h h1 P 1 h+ h P h ξ h11 h 1 P 1 h. 3c Ths s formally the same optmzaton problem as that for the pont-to-pont MISO wth PerPC n 17 and thus the optmal powers are gven by P 1 h = P h = 1 h1 1 h1 > 0, 33 1 h > We next need to verfy that, for these optmzng powers, we satsfy 3c; by dong so, we conclude that P1 h, P h, ρ h =eq.33, eq.34, 1 s optmal when { R 4 := eq.5 holds, h 11 h 1 + h, > 1 }. 3 On the Optmalty 0 < ρh < 1: As mentoned earler, f P 1 h P h =0n 4 then trvally ρ h =0s optmal and thus under the assumpton that ρ h > 0 we only need to consder P 1 h P h > 0. When P 1 h > 0, P h > 0 and 0 < ρ h < 1 the system of equatons n 6 does not seem to have a closed form soluton. Therefore n R 5 := {eq.5 holds, R 1 R R 3 R 4 R 5 c } the optmal power allocaton must be found numercally. REFERENCES [1] D. Maamar, D. Tunnett, and N. Devroye, The capacty of the ergodc MISO channel wth per-antenna power constrant and an applcaton to the fadng cogntve nterference channel, n Proc. IEEE Int. Symp. Inf. Theory, Honolulu, HI, USA, Jun. 014, pp [] A. Goldsmth, S. Jafar, I. Marc, and S. Srnvasa, Breakng spectrum grdlock wth cogntve rados: An nformaton theoretc perspectve, Proc. IEEE, vol. 97, no. 5, pp , May 009. [3] L. Sankar, X. Shang, and V. Poor, Ergodc fadng nterference channels: Sum-capacty and separablty, IEEE Trans. Inf. Theory, vol. 57, no. 5, pp , May 011. [4] R. A. Tannous and A. Nosratna, Cogntve rado protocols based on explotng hybrd ARQ retransmssons, IEEE Trans. Wreless Commun., vol. 9, no. 9, pp , Sep [5] V. Annapureddy, A. El Gamal, and V. V. Veeravall, Degree of freedom of nterference channels wth CoMP transmsson and recepton, IEEE Trans. Inf. Theory, vol. 58, no. 9, pp , Sep. 01. [6] A. El Gamal and V. V. Veeravall, Dynamc nterference management, n Proc. Aslomar Conf. Sgnals, Syst. Comput., Pacfc Grove, CA, USA, Nov. 013, pp [7] D. Maamar, D. Tunnett, and N. Devroye, Approxmate sum-capacty of K-user cogntve nterference channels wth cumulatve message sharng, IEEE J. Select. Areas Commun., vol. 3, no. 3, pp , Mar [8] D. Tunnett, Gaussan fadng nterference channels: Power control, n Proc. Aslomar Conf. Sgnals, Syst. Comput., Monterrey, CA, USA, Oct. 008, pp [9] V. R. Cadambe and S. A. Jafar, Parallel Gaussan nterference channels are not always separable, IEEE Trans. Inf. Theory, vol. 55, no. 9, pp , Sep [10] D. Tse and S. V. Hanly, Multaccess fadng channels. I. Polymatrod structure, optmal resource allocaton and throughput capactes, IEEE Trans. Inf. Theory, vol. 44, no. 7, pp , Nov [11] L. L and A. Goldsmth, Capacty and optmal resource allocaton for fadng broadcast channels Part I: Ergodc capacty, IEEE Trans. Inf. Theory, vol. 47, no. 3, pp , Mar [1] A. Ghasem and E. S. Sousa, Fundamental lmts of spectrum-sharng n fadng envronments, IEEE Trans. Wreless Commun., vol. 6, no., pp , Feb [13] X. Kang, Y.-C. Lang, A. Nallanathan, H. K. Garg, and R. Zhang, Optmal power allocaton for fadng channels n cogntve rado networks: Ergodc capacty and outage capacty, IEEE Trans. Wreless Commun., vol. 8, no., pp , Feb [14] R. Zhang, S. Cu, and Y.-C. Lang, On ergodc sum capacty of fadng cogntve multple-access and broadcast channels, IEEE Trans. Inf. Theory, vol. 55, no. 11, pp , Nov [15] M. Vu, MISO capacty wth per-antenna power constrant, IEEE Trans. Commun., vol. 59, no. 5, pp , May 011.

12 80 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO., FEBRUARY 015 [16] N. Devroye, P. Mtran, and V. Tarokh, Achevable rates n cogntve rado channels, IEEE Trans. Inf. Theory, vol. 5, no. 5, pp , May 006. [17] A. Jovcc and P. Vswanath, Cogntve rado: An nformaton-theoretc perspectve, IEEE Trans. Inf. Theory, vol. 55, no. 9, pp , Sep [18] S. Rn, D. Tunnett, and N. Devroye, New nner and outer bounds for the dscrete memoryless cogntve nterference channel and some new capacty results, IEEE Trans. Inf. Theory, vol. 57, no. 7, pp , Jul [19] S. Rn, D. Tunnett, and N. Devroye, Inner and outer bounds for the Gaussan cogntve nterference channel and new capacty results, IEEE Trans. Inf. Theory, vol. 58, no., pp , Feb. 01. [0] T. Cover and J. Thomas, Elements of Informaton Theory: Second Edton. Hoboken, NJ, USA: Wley, 006. [1] M. Costa, Wrtng on drty paper, IEEE Trans. Inf. Theory, vol. IT-9, no. 3, pp , May [] Y. Zhu and M. Vu, Iteratve mode-droppng for the sum capacty of MIMO-MAC wth per-antenna power constrant, IEEE Trans. Commun., vol. 60, no. 9, pp , Sep. 01. [3] D. Tunnett, On the capacty of the AWGN MIMO channel under perantenna power constrants, n Proc. IEEE Int. Conf. Commun., Sydney, Australa, Jun. 014, pp [4] Z. P, Optmal transmtter beamformng wth per-antenna power constrants, n Proc. IEEE Int. Conf. Commun., Ottawa, ON, Canada, Jun. 01, pp [5] R. Knopp and P. A. Humblet, Informaton capacty and power control n sngle-cell multuser communcatons, n Proc. IEEE Int. Conf. Commun., Seattle, WA, USA, Jun. 1995, vol. 1, pp Dana Maamar receved the Bachelor of Scence degree wth honors n electrcal engneerng and the Master of Scence degree wth honors n electrcal engneerng from the Unversty of Balamand, Al Koura, Lebanon, n 009 and 011, respectvely. She s currently workng toward the Ph.D. degree wth the Unversty of Illnos at Chcago, IL, USA. Her current research focuses on mult-user nformaton theory and ts applcatons to cogntve rado channels. Natasha Devroye receved the B.Eng. Hons. degree n electrcal engneerng from McGll Unversty, Montreal, QC, Canada, n 001 and the M.Sc and Ph.D. degrees n engneerng scences from Harvard Unversty, Cambrdge, MA, USA, n 003 and 007, respectvely. She has been an Assstant Professor wth the Department of Electrcal and Computer Engneerng, Unversty of Illnos at Chcago UIC, IL, USA, snce January 009. From July 007 untl July 008, she was a Lecturer at Harvard Unversty. Her research focuses on multuser nformaton theory and applcatons to cogntve and software-defned rado, radar, and relay and two-way communcaton networks. She s an Edtor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and the IEEE JOURNAL OF SELECTED AREAS OF COMMUNICATIONS Cogntve Rado Seres. She was a recpent of the NSF CAREER Award n 011 and was named UIC s Researcher of the Year n the Rsng Star category n 01. Danela Tunnett receved the M.S. degree n telecommuncaton engneerng from the Poltecnco d Torno, Turn, Italy, n 1998 and the Ph.D. degree n electrcal engneerng from ENST/Telecom ParsTech, Pars, France wth work done at the Eurecom Insttute, Sopha Antpols, France, n 00. From 00 to 004, she was a Postdoctoral Research Assocate wth the School of Communcaton and Computer Scence, EPFL/Swss Federal Insttute of Technology n Lausanne, Lausanne, Swtzerland. Snce January 005, she has been wth the Department of Electrcal and Computer Engneerng, Unversty of Illnos at Chcago, Chcago, IL, USA, where she currently s an Assocate Professor. Her research nterests are n the ultmate performance lmts of wreless nterference networks, wth specal emphass on cognton and user cooperaton. She was the Edtor-n-Chef of the IEEE Informaton Theory Socety Newsletter from 006 to 008, an Assocate Edtor of the IEEE COMMUNICATION LETTERS from 006 to 009, and an Edtor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS from 009 to 014. She currently s an Assocate Edtor of the IEEE TRANSACTIONS ON INFORMATION THEORY. She regularly serves on the techncal program commttees of IEEE workshops and conferences. She was the Communcaton Theory Symposum Co-Char of the 010 IEEE Internatonal Conference on Communcatons ICC 010. She was a recpent of the Best Student Paper Award at the European Wreless Conference n 00 and the NSF CAREER Award n 007.