Distributed Power Control for Interference-Limited Cooperative Relay Networks

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1 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2009 proceedngs Dstrbuted Power Control for Interference-Lmted Cooperatve Relay Networks Sheng Zhou, Hongda Xao, and Zhsheng Nu Tsnghua Natonal Laboratory for Informaton Scence and Technology Dept. of Electronc Engneerng, Tsnghua Unversty, Bejng , Chna zhouc@mals.tsnghua.edu.cn, xhd05@mals.tsnghua.edu.cn, nuzhs@tsnghua.edu.cn Abstract In ths paper, a dstrbuted power control algorthm s proposed for wreless relay networks n nterference-lmted envronments. The objectve s to mnmze the total transmsson power whle satsfyng the sgnal-to-nterference-plus-nose rato (SINR) requrements. Two forwardng technques,.e., decodeand-forward (DF) and amplfy-and-forward (AF), are consdered. The proposed algorthm only requres locally measured SINR on the relay nodes (RNs) and the destnaton nodes (DNs), based on whch each cooperaton unt (defned as one source node (SN) and DN par wth the RN assocated to t) teratvely updates the transmsson power of the SN and the RN by solvng a local optmzaton problem. We prove that the convergence s guaranteed when the parameters adopted n the algorthm are suffcently large, and then a parameter adjustng method s also desgned. Smulaton results ndcate that the proposed algorthm converges fast and leads to only 7% more power consumpton than the optmal power allocaton n the consdered scenaros. It s also shown that even n nterference-lmted envronments, relayng can stll mprove system performance substantally n terms of outage and power consumpton. 1 I. INTRODUCTION Cooperatve transmsson has been recevng much attentons as a promsng technology for the future wreless networks [1] [2]. Snce relay nodes (RNs) can provde addtonal spatal dversty, the advantage of relayng s twofold: reducng sgnal transmsson power and mprovng data rate. Extensve researches have been made on resource allocaton for cooperatve relay communcatons, such as power control to guarantee outage performance [4], relay node selecton combned wth power control to enhance capacty [5], etc. All these efforts have helped understandng the benefts of cooperatve relays and nspred usng relayng n practcal systems, for nstance, mult-hop cellular networks [6]. However, most of the exstng work s based on solated relay transmsson,.e., only one source node (SN) and destnaton node (DN) par s presented, or transmssons are orthogonal, where nterference s not taken nto account. On the other hand, to adopt relayng n practcal systems, such as wreless cellular networks or wreless sensor networks, nterference s an nevtable ssue to deal wth. In these systems, the nterference exsts not only on SN to DN lnks, 1 Ths work has been partally sponsored by the Natonal Basc Research Program of Chna (973 Program: 2007CB310607); by NSFC/Hong Kong Research Grants Councl Jont Research Scheme under Grant ; and by the Internatonal S&T Cooperaton Program of Chna (ISCP) (No. 2008DFA12100). but may also exst on SN to RN lnks and RN to DN lnks. In [8], the authors show that, the performance of relayng n large-scale wreless networks s penalzed by the elevated level of nterference nduced by the RNs. Hence, a nature problem has arsen: How much gan can we get from relayng n nterference-lmted envronments [3], and how can we desgn resource allocaton schemes to acheve the gan? One way to nvestgate ths problem s to consder power control. As a basc way to facltate spatal reuse, power control plays a fundamental role n system desgn and performance evaluaton. There are few papers addressng power control for nterference-lmted relay networks, and most of them provde centralzed solutons. In [8], a channel allocaton mechansm s proposed to mtgate the mpact of nterference, where the power control s n fact bnary (on-off) on the allocated channels. The authors n [6] desgn power control and RN selecton schemes for multhop cellular networks consderng nter-cell nterference, and based on the assumpton that the second hop s suffcently good, the power control s smlar to the ones for sngle-hop scenaro descrbed n [10]. The most related work to ours s Ref. [7], whch studes power allocaton problem for the uplnk n cooperatve CDMA networks. Interference nduced by all the nodes n the system (ncludng RNs and SNs) s consdered, and a type of complementary geometrc programmng (GP) method [9] s used to solve the power optmzaton problem, whch requres global channel state nformaton (CSI) thus can only be used n a centralzed way. Moreover, when the number of smultaneous transmssons ncreases, the GP method wll requre very long computatonal tme. The dffculty of collectng global CSI and the hgh realzaton complexty lmt the usage of centralzed solutons n practcal systems, such as cellular systems when multple Base Statons (BSs) are consdered. Therefore, effectve dstrbuted power control usng local measurements s of more practcal sgnfcance. In ths paper, we propose a dstrbuted yet effectve power control algorthm for nterference-lmted cooperatve relay networks. In partcular, the algorthm ams to mnmze the total transmsson power subject to the sgnal-to-nterferenceplus-nose rato (SINR) requrements. Each cooperaton unt (defned as one SN-DN par wth the RN assocated to t) utlzes the locally measured SINR to solve a smple optmzaton problem, of whch the soluton can be wrtten n close-form. The algorthm requres low computatonal over /09/$ IEEE Authorzed lcensed use lmted to: Tsnghua Unversty Lbrary. 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2 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2009 proceedngs Cooperaton unt Cooperaton unt j SN RN DN Fg. 1. G r, G jr, G d, G r, d G jd, System Model G r j, d Transmsson on CH1 Interference on CH1 Transmsson on CH2 Interference on CH2 head, and together wth the dstrbuted feature, t can be easly adopted n practcal large-scale wreless networks. The remanng of the paper s organzed as follows. The system model s ntroduced n Secton II. In Secton III, the proposed algorthm s descrbed and ts convergence property s studed, based on whch the parameter adjustment method to guarantee convergence s also detaled. Numercal results are provded and the performance of the proposed algorthm s evaluated n Secton IV. Fnally, Secton V concludes the paper. II. SYSTEM MODEL AND PROBLEM FORMULATION We consder a 2-hop relay network. As shown n Fg.1, there are N SN-DN pars, and each par s asssted by one RN to form a cooperaton unt (CU). Two orthogonal channels (tme slots or frequency bands, etc.) are avalable: the SNs transmt on channel 1 and the RNs transmt on channel 2 (.e., out-of-band relay). The RNs can receve on channel 1, whle the DNs can combne the sgnals receved on both two channels. Although the transmssons from the SNs and RNS are separated on the two channels, the smultaneous transmssons on channel 1 from dfferent SNs wll cause nterference to each other, and so do the RNs on channel 2. On channel 1, we denote G j,d as the channel gan from SN j to DN, and G j,r as the channel gan from SN j to RN. On channel 2 we denote G rj,d as the channel gan from RN j to DN. For smplcty, we omt the channel ndces on the notatons of the channel gans. The receved SINR at DN on channel 1 s gven by γ sd, G,d P s N j1 G j,d P sj G,d P s + η, (1) where P s s the transmsson power of SN, and η s the power of the background addtve whte Gaussan nose. Smlarly, the receved SINR at RN s γ sr, G,r P s N j1 G j,r P sj G,r P s + η. (2) We further denote P r as the transmsson power of RN, whch wll nterfere wth the transmssons of other RNs. The receved SINR of the forwarded sgnal from RN at DN on channel 2 s gven by γ rd, G r,d P r N j1 G r j,d P rj G r,d P r + η. (3) At the DNs, maxmum rato combnng (MRC) s used to combne the sgnals receved from the two channels. The equvalent SINR after MRC s the sum of the drect lnk SINR and the relay lnk SINR. For decode-and-forward (DF) relayng, the equvalent SINR s γ DF γ sd, + mn{γ sr,,γ rd, }, (4) where the second term means that the qualty of the relay lnk s constraned by the qualty of the SN-RN and RN-DN lnks. For amplfy-and-forward (AF) relayng, the RN amplfes the receved sgnal together wth nterference and nose and forwards to the DN. The equvalent relay lnk SINR n terms of the transmtted sgnal from the SN s gven by γ eq γ sr,γ rd, γ sr,+γ rd, +1 rd,. Hence, the equvalent SINR after combnng s γ AF γ sr, γ rd, γ sd, + γ sr, + γ rd, +1. (5) In order to guarantee successful transmsson, the combned SINR should be larger than a threshold: γ.theresalsoa maxmum transmsson power lmt P max for each SN/RN. Here the power control problem can be descrbed as to mnmze the total transmsson power on the SNs and RNs subject to the combned SINR constrants. For DF relayng, n order for the RN to decode the sgnal from the SN successfully, we also have γ sr, γ. Then the power optmzaton problem can be formulated as follows for DF relayng: N mnmze (P s + P r ) 1 subject to γ sd, + γ rd, γ,, max{γ sd,,γ sr, } γ, 0 P s P max,, 0 P r P max,, where the the second constrant ndcates that f the drect lnk satsfes the SINR requrement, the RN does not need to provde help. Otherwse, γ sr, γ,soγ rd, can smply be no larger than γ sr,, then from (4), we get the frst constrant. For AF relayng, the optmzaton problem can be formulated as: N mnmze (P s + P r ) 1 subject to γ sd, + γsr,γ rd, γ sr,+γ rd, +1 γ, 0 P s P max,, 0 P r P max,. Due to the exstence of the frst constrant n (6) and (7), t can been seen that they are not convex optmzaton problems [7], therefore hard to be effcently solved. In the next secton, a dstrbuted algorthm to solve the above power control problems s descrbed.,, (6) (7) Authorzed lcensed use lmted to: Tsnghua Unversty Lbrary. Downloaded on September 26, 2009 at 10:34 from IEEE Xplore. Restrctons apply.

3 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2009 proceedngs III. PROPOSED DISTRIBUTED ALGORITHM 0.8 In ths secton, we provde a dstrbuted power control algorthm for both DF and AF relayng. In our work, dstrbuted means that sgnalng exsts only between the RN, DN and the SN that belongs to one CU. No sgnalng s exchanged between dfferent CUs. Ths assumpton s reasonable because the DN always needs to feedback the SINR nformaton to ts RN and SN to accomplsh power control (for example, the uplnk power control n cellular networks), meanwhle the sgnalng can make use of the feedback channel. We frst make the followng notatons * P s P r ω P s +P r a b c G,d γ sd, N j1 G j,d P sj G,d P s +η P s, G,r N j1 Gj,r Ps j G,r Ps +η γsr, P s, G r,d N j1 G r j,d P rj G r,d P r +η γ rd, P r. (8) P s Fg. 2. Relaton between P s, P r and the object functon n (9) for AF relayng Snce γ sd,, γ sr, and γ rd, can be locally measured by RN /DN, alla, b and c can be acqured locally by each CU. We further ntroduce a power control weght vector ω {ω }, whch actually represents the dfferent mpacts on the system power consumpton from P s and P r. The proposed teratve dstrbuted power control algorthm can be descrbed n a slot (ndexed by t) by slot manner: Step 0 Set all P (0) s, and P (0) r, wth an ntal value, eg. zero, and set t1; Step 1 Each CU solves a local optmzaton problem based on a (t 1), b (t 1) the last tme slot and c (t 1) measured at the end of mn ω P s + P r s. t. 0 P s P max, 0 P r P max, (DF) a (t 1) P s + c (t 1) P r γ, (AF) a (t 1) P s + b(t 1) P s c (t 1) P r b (t 1) P s +c (t 1) P r +1 γ. (9) Notce that for DF relayng, the SINR constrant should also nclude the second constrant n (6): max{a (t 1) P s,b (t 1) P s } γ. Suppose the soluton to (9) s P s (t) and P r (t), then ω s also updated; Step 2 SN transmts wth P s (t) and RN transmts wth P r (t). Increase t by 1 and go back to Step 1 untl converges. The convergence crtera s (t) ( P s P s (t 1) + P r (t) P r (t 1) ) <ɛ, where ɛ s the error tolerance for ext condton. Next, the close-form soluton to (9) wll be detaled for DF and AF relayng respectvely, and the method of tunng ω wll be provded. We wll replace a (t 1), b (t 1) and c (t 1) by a, b and c respectvely wthout ambguty. A. Detaled Algorthm for DF Relayng For DF relayng, the local optmzaton problem (9) s a smple lnear programmng problem. We change the SINR constrant n (9) to an equalty, and the optmzaton problem remans the same. Then P r can be represented by a functon of P s (f b P s γ holds): P s (t) P r max{0, (γ a P s )/c }. (10) Hence (9) can be easly solved as: mn{p max, γ a }, r 0; f a >b s r s cp (t) r mn{p max, γ a }, mn{p max, (b a)γ b c }; f a b and ω c a mn{p max, γ a }, P r (t) mn{p max, γ (t) ap s c }. f a b and ω c <a (11) In the above explanaton, under the second condton, we actually get P r (t) frst then decde P s (t), and vce verse under the thrd condton. Notce that (11) ncludes the power allocaton results when the local problem s nfeasble, whch leads to (P s (t),p r (t) )(P max, 0) or (P max,p max ), whle the SINR requrement s not satsfed n these crcumstances. B. Detaled Algorthm for AF Relayng For AF relayng, we also change the SINR constrant n (9) to an equalty, and the optmzaton problem remans the same. Then P r can be represented by a functon of P s (f P s γ /(a + b ) holds): where we defne f(x) as P r max{0,f(p s )}, (12) f(x) (1 + b x)(γ a x) c (b x + a x γ ), for x> γ. (13) a + b The relaton between P s and P r s shown n Fg. 2. Furthermore, we can easly verfy that f (x) > 0 n ts doman, thus the local optmzaton problem s now a convex optmzaton problem. If we defne Ps as the soluton to the formula f (x)+ω 0, (14) Authorzed lcensed use lmted to: Tsnghua Unversty Lbrary. Downloaded on September 26, 2009 at 10:34 from IEEE Xplore. Restrctons apply.

4 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2009 proceedngs whch leads to (as shown n Fg.2) [ ] Ps 1 γ (a b + b 2 + γ b2 ) + γ. (15) a + b c ω (a + b ) a b Note that Ps may not exst when c ω (a + b ) a b 0. Then the soluton to (9) wthout maxmum power constrant s gven by s r { γ a, f P s γ a or P s does not exst P s, f P s < γ a f( s ). (16) If P max γ /(a + b ), then the local problem s nfeasble; otherwse, f any one of P s (t) and P r (t) volates the P max constrant, t wll be set as P max, and the other one wll be decded by (12) or ts nverse. Moreover, f the result volates the power constrant agan, we wll set (P s (t),p r (t) ) (P max,p max ), but the SINR requrement s not satsfed. C. Adjustng Parameter ω As mentoned prevously, weght parameter ω actually represents the dfferent mpacts on the system power consumpton from P s and P r. It has two functons: (1) To mprove the performance of the proposed algorthm. We note that the proposed algorthm s sub-optmal, however, by balancng the mpacts from P s and P r on the system power consumpton, the result can be very close to the global optmum. Ths can be done by optmzng ω for each CU; (2) To guarantee convergence. For an arbtrary ω, the proposed algorthm s not guaranteed to converge. However, by approprately tunng ω, we can actually keep the algorthm converge, whch s descrbed n the followng proposton: Proposton 1: If the power control problem s feasble, there exsts an ω 0 2, for any ω ω, the proposed algorthm wth parameter ω wll converge to an equlbrum. See Appendx for the proof. The above proposton ndcates that, when the proposed algorthm does not converge, by ncreasng the elements of ω, the algorthm wll fnally converge. Thus, the dynamc adaptaton of ω should detect oscllatons and approprately ncrease each ω by evaluatng the mpact on the system transmsson power from P s. The optmal choce of ω for each CU depends on the nterference structure of the system, thus requres global channel nformaton. In fact, determnng optmal ω s of the same dffculty wth gettng the global optmum of the orgnal optmzaton problem (6) and (7), whch s very hard. Hence we adopt a heurstc way to adapt ω. Intally, for each CU, wesetω (0) Then durng each teraton, ω can be updated as ω (t+1) ω (t) + α (t) s as the ntal value. P (t 1) s, (17) where α (t) > 0 s a stepsze. We set α (t) α 0 / (t t 0 ),for some constant α 0 > 0 and postve nteger t 0, and f t t 0, α (t) 0. The ntuton behnd (17) s descrbed as follows: 2 represents the component-wse nequalty between vectors. TABLE I SIMULATION PARAMETERS Parameter Value Parameter Value Background Nose η SINR Requrement γ 5 Maxmum Power P max 1 Step Constant α 0 5 Tme shft t 0 1 Intal ω (0) 1 Durng the teratons, the larger the varance of P s s, the larger mpact of t on the total system transmsson power, hence ω s ncreased proportonally to the varance of P s between the current tme slot and the last tme slot. In the proof of proposton 1, the choce of ω s n a most conservatve way. Actually, ω depends on the proporton of the channel gan dvded by the nterference level on the SN- DN lnk to that on the RN-DN lnk. The proporton s generally not large and can even be less than one, because most of the tme the channel and nterference condton on the RN-DN lnk s better n real systems due to two reasons: 1) More relable rado technque s used n these lnks, and RNs are generally closer to the DNs/BSs; 2) RNs are not fully utlzed as wll be shown n the numercal results, hence the nterference level on the RN-DN lnks s relatvely low. IV. NUMERICAL RESULTS AND DISCUSSIONS In the smulatons, wreless lnks are unformly dstrbuted over a square feld wth dmenson D D. The path gan G,j between any two nodes (SN, RN or DN) s modeled as G,j 1, where d d 4,j s the dstance between node,j and node j. Other smulaton parameters are lsted n Table I. In each run, we frst randomly generate N SNs and ther correspondng DNs n ths area, and the dstance between any SN and ts DN s upper hounded by a maxmum dstance D M. We then generate MN canddate RNs, denoted as set C, and the selecton of the RN for each SN follows a smple the best worst [7] crtera: each SN selects r arg max j C mn(g,j,g j,d ). Ths crtera ams to balance the two hops and can easly be mplemented. Durng the smulatons, f the power control problem s not feasble, then there s at least one CU whose combned SINR requrement s not satsfed, the one wth the lowest SINR s shuttng off and the algorthm s restarted. Ths s repeated untl the algorthm converges and the SINR requrement of all CUs are satsfed. Whle more complcated RN selecton and feasble set selecton schemes can mprove the performance, ths s at a prce of ncreasng mplementaton overhead and processng tme, and s beyond the scope of ths paper. We frst demonstrate the convergence of the proposed scheme. In ths set of smulatons, D 1000, D M 50 and M 6. We show the cumulatve dstrbuton functon (CDF) of the teraton tme needed for the algorthm under dfferent condtons n Fg. 3. When the error tolerance s tght as ɛ : Wth large number of CUs N 30, the average number of teratons needed s less than 30 for AF and 40 for DF; Wth N 10, the average number of teratons needed s less than 10, whch s very small. Generally, the Authorzed lcensed use lmted to: Tsnghua Unversty Lbrary. Downloaded on September 26, 2009 at 10:34 from IEEE Xplore. Restrctons apply.

5 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2009 proceedngs Cumulatve dstrbuton functon DF, N30, ε10 6 AF, N30, ε10 6 DF, N30, ε10 4 AF, N30, ε10 4 DF, N10, ε10 6 AF, N10, ε Number of teratons needed Average number of feasble CUs (SN DN pars) DF proposed M1 DF proposed M6 DF proposed M12 AF proposed M1 AF proposed M6 AF proposed M12 No relayng Number of total SN DN pars, N Fg. 3. CDF of the number of teraton tme slots Fg. 5. The average number of feasble CUs v.s. N Average transmsson power for each CU (SN DN par) DF proposed AF proposed No relayng AF SC GP Average number of feasble CUs (SN DN pars) Fg. 4. Comparson of average transmsson power teraton tme for AF relayng s less than DF relayng. 3 When ɛ whch s stll acceptably small, the teraton tme s substantally reduced, especally for AF relayng. We notce that the teraton tme wll be long for some sckly generated topologes, n whch the nterference level on RN-DN lnks s very hgh. The teraton tme can be reduced by settng α(t) α 0 as a constant, but t wll degrade the performance of power control. Snce ths stuaton rarely happens n practcal systems (lke uplnk n cellular systems, the RNs are generally close to the BSs and undergoes low nterference level), we stll use the step sze descrbed n Secton III-C. Fg. 4 shows the beneft of usng relayng n terms of savng transmsson power. In ths set of smulatons, D 300, D M 70, the number of canddate RNs s fxed as 30. We change N from 2 to 8, hence t s a crowded scenaro wth severe nterference, whch leads to hgh outage. It s shown that even under ths crcumstance, the transmsson power wth relayng s sgnfcantly reduced comparng to the norelayng case, and from the flat slope of the curves, relayng shows robustness to the ncreasng of the nterference level. In other words, relayng can keep the power consumpton of 3 Ths s due to the strct convexty of the local vrtual power optmzaton problem for AF relayng as descrbed n the Appendx. each CU even when the number of smultaneous transmssons s growng. We also compare our proposed algorthm wth the successve convex approxmaton GP (SC-GP) algorthm adopted n [7] and [9], whch s shown to be able to reach the global optmum wth hgh probablty. From Fg. 4, we can see that the proposed scheme only consumes about 3% to 7% more power than SC-GP does for AF relayng. 4 The DF relayng consdered s dfferent than the one n [7] because of the second constrant we use n (6), whch means that n our algorthm, relayng s not compulsve f the SN-RN lnk s bad. However, problem (6) can not be solved by SC-GP, so only the AF example gven by SC-GP s provded. It s also shown that the DF relayng s better than AF relayng 5, because f relayng s benefcal, from (4) and (5), γ AF γ DF holds, thus DF relayng requres less power. We then study the outage performance of relayng n Fg. 5. In ths set of smulatons, D 1000, D m 70, and the number of canddate RNs per SN-DN par s M 1, 6 and 12 respectvely. As the number of smultaneous transmssons ncreases, both AF and DF relayng outperforms no-relayng sgnfcantly. Even wth M 1, the performance gan s notceable. It addton, the performance of DF and AF relayng s very close, and only when M 1, AF relayng outperforms DF relayng slghtly because the probablty that the recevng SINR at the RNs can not satsfy γ s hgh. On the other hand, ths dsadvantage can be compensated by the lower nterference level n DF relayng as a result of lower transmsson power shown n Fg. 4. Besdes what we have delvered n Fg. 5, n the smulatons, we have also observed that, although each SN-DN par s assgned wth an RN, the usage (the percentage of RNs that wth non-zero transmsson power) s not 100%. For example, when N 70, M 6,the usage of RN s only 38.2% for DF, and 44.5% for AF. Ths ndcates that wth more advanced RN selecton scheme, the performance can stll be guaranteed even when the number 4 Because the SC-GP requres a feasble ntal power allocaton, we use the results obtaned by our proposed algorthm as the ntal value for SC-GP, leadng to the same outage probablty. 5 In [7], DF performances rather bad because relayng s compulsve even f t s not benefcal, and the RN must decode the sgnal from the SN. Authorzed lcensed use lmted to: Tsnghua Unversty Lbrary. Downloaded on September 26, 2009 at 10:34 from IEEE Xplore. Restrctons apply.

6 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the IEEE ICC 2009 proceedngs of canddate RN s lmted. Thus the desgn of RN selecton scheme n nterference-lmted envronments s valuable and yet to be further nvestgated. V. CONCLUSION We have proposed a dstrbuted power control algorthm to mnmze the total system transmsson power subject to the SINR requrements n nterference-lmted envronments. The proposed algorthm only requres locally measured SINR, and the computatonal overhead s low. Smulaton results have shown that the proposed algorthm performs close to the optmal soluton and the benefts of relayng under nterference-lmted condtons are demonstrated. The proposed power control algorthm can be utlzed n multple ways, such as evaluatng the effectveness of relayng n multcell envronment where co-channel nterference exsts among cells. Moreover, as the future work, nterference-aware RN selecton scheme should be consdered together wth the power control algorthm to effectvely explot the spatal dversty that relayng provdes. ACKNOWLEDGMENT The authors would lke to express ther sncere thanks to Htach R&D Headquarter for the contnuous supports. Furthermore, the authors gratefully acknowledge the support from the R&D Center, Chna Moble Communcatons Corporaton. REFERENCES [1] J. N. Lanemen, D. N. C. Tse and G. W. Wornell, Cooperatve dversty n wreless networks: effcent protocols and outage behavor, IEEE Trans. Inform. Theory, vol. 50, no. 12, pp , Dec [2] A. Sendonars, E. Erkp and B. Aazhang, User cooperaton dversty, Part I: System descrpton, IEEE Transactons on Communcatons, vol. 51, no. 11, pp , Nov [3] V. Morgenshtern and H. Bölcske, On the value of cooperaton n nterference relay networks, n CD Record Allerton Conference, Dec [4] M. O. Hasna and M.-S. Aloun, Optmal power allocaton for relayed transmssons over Raylegh fadng channels, Proc. IEEE VTC 2003, vol. 4, pp , Jeju, Korea, Apr [5] B. Zhang, Z. Han, K. J. R. Lu, Dstrbuted relay selecton and power control for multuser cooperatve communcaton networks usng buyer/seller game, Proc. IEEE INFOCOM 2007, pp , Anchorage, AK, May [6] L. Le, E. Hossan, Multhop cellular networks: potental gans, research challenges, and a resource allocaton framework, IEEE Communcatons Magazne, vol. 45, no. 9, pp , Sep [7] B. Wang and D. Zhao, Optmal power dstrbuton for uplnk channel n a cooperatve wreless CDMA network, Proc. IEEE ICC 2008, pp , Bejng, Chna, May [8] Y. Zhu and H. Zheng, Understandng the mpact of nterference on collaboratve relays, IEEE Transactons on Moble Computng, vol. 7., no. 6., pp , Jun [9] M. Chang, C. W. Tan, D. P. Palomar, D. O Nell and D. Julan, Power control by geometrc programmng, IEEE Transactons on Wreless Communcatons, vol. 6, No. 7, pp , Jul [10] R. Yates, A framework for uplnk power control n cellular rado systems, IEEE Journal on selected Areas n Communcatons, vol. 13, no. 7, pp , Sep [11] W. Y. Zhang, Game Theory and Informaton Economcs, n Chnese. Shangha Peoples Publshng House, Shangha, Chna, APPENDIX Proof: (Proof of Proposton 1) Because we assume that the power control problem s feasble, the volaton to the maxmum power constrant s not consdered. Frst, for DF relayng, from (11), assume that for each CU, ω > a /c always exsts, then the soluton wll be P s (t) γ /a,p r (t) 0, for a > b or P s (t) γ /b,p r (t) (b a )γ /(b c ), for a < b. Now our proposed algorthm s equvalent to a two-stage power control problem: The frst stage s on channel 1, each SN tres to acheve max{γ sd,,γ sr, } γ ; The second stage s on channel 2, each RN tres to fulfll the remanng SINR gap max{γ γ sd,, 0}, based on the result of the frst stage. It can be easly shown that the update of P s (t) and P r (t) n each teraton can be categorzed as a standard nterference functon [10] respectvely, hence the convergence to a fxed pont s guaranteed as proved n [10]. Moreover, f we let: ω G,d ( N j1,j G r j,d P max + η). (18) ηg r,d Snce ω >a /c always holds (from the defnton n (8) and the maxmum power constrants), for any ω >ω,the algorthm wll converge. Second, for AF relayng, we construct vrtual powers for each CU : P s and P r, and they do not have non-negatve constrants,.e., the vrtual powers can take negatve values. Actually, each CU frst solves a local optmzaton problem n terms of the vrtual powers: mnmze ω P s + P r subject to P r f(p s ), (19) where f(x) s defned n (13). Solvng ths problem s actually solvng the formula (14), and the soluton s gven by (15). Assume that the soluton to (19) always exsts, say, P s and P r, then each CU sets the transmsson power as mn{ P s,γ /a } and P r (t) max{0, P r }. Because s f(p s ) s convex and wth the assumpton that the soluton to (19) always exsts, by Debreu s theorem [11], the proposed algorthm wll converge to an equlbrum. Now t remans to guarantee the exstence of the soluton to (19), whch requres that ω > ab c (a +b ) (from (15)). Smlar to DF relayng, f we set ω the same as (18), snce ω > a c > ab c (a +b ) always holds (from the defnton n (8) and the maxmum power constrants), for any ω >ω, the algorthm wll converge to an equlbrum. Authorzed lcensed use lmted to: Tsnghua Unversty Lbrary. Downloaded on September 26, 2009 at 10:34 from IEEE Xplore. Restrctons apply.