Optimizing Power Allocation in Interference Channels Using D.C. Programming

Transcription

1 Otimizing Power Allocation in Interference Channels Using D.C. Programming Hussein Al-Shatri, Tobias Weber To cite this version: Hussein Al-Shatri, Tobias Weber. Otimizing Power Allocation in Interference Channels Using D.C. Programming. WiOt 10: odeling Otimization in obile, Ad Hoc, Wireless Networks, ay 2010, Avignon, France , <inria > HAL Id: inria htts://hal.inria.fr/inria Submitted on 13 Jul 2010 HAL is a multi-discilinary oen access archive for the deosit dissemination of scientific research documents, whether they are ublished or not. The documents may come from teaching research institutions in France or abroad, or from ublic or rivate research centers. L archive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recherche, ubliés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires ublics ou rivés.

2 RAWNET 2010 Otimizing Power Allocation in Interference Channels Using D.C. Programming Hussein Al-Shatri Tobias Weber Institute of Communications Engineering University of Rostock Richard-Wagner-Str. 31, Rostock, Germany Telehone:49) , Fax:49) Abstract Power allocation is a romising aroach for otimizing the erformance of mobile radio systems in interference channels. In the resent aer, the non-convex objective function of the ower allocation roblem aiming at maximizing the sum rate with a total ower constraint is reformulated as a difference of two concave functions. A global otimum ower allocation isfoundbyalyingabranchboundbasedalgorithmto the new formulation. The algorithm basically slits the feasible region consecutively into subregions where for every subregion the objective function is uer lower bounded. For a certain artition of the feasible region, a ower allocation corresonding to the highest lower bound which is uer bounded by the highest uer bound with some insignificant difference is found as the global otimum. A convex maximization formulation of the otimization roblem with a iecewise linearly outer aroximated feasible region is essentially alied for finding an uer bound which only requires solving a linear rogram roblem. The simulation results show a significant imrovement in the sum rate of the roosed algorithm over the conventional subotimal techniques. I. INTRODUCTION Interference is a dominant source of erformance degradations in mobile radio systems. If the nodes communicate autonomously through a shared channel, i.e., there is no cooeration among nodes, the interference channel can be considered as a system level model1]. In the interference channel, it is assumed that a number of transmitter-receiver airs communicate with each other through a shared medium where any transmission from a transmitter would not just result inausefulsignalatitscorresondingreceiverbutalsoinan interference signal at all other receivers. Power allocation lays a key role in imroving the system erformance in interference channels. A smart ower allocation resulting in minimizing the received interference thus maximizing the system sum rate is required. If the interference is treated as noise, the ower allocation otimization roblem aimingatmaximizingthesumratewithatotalowerconstraint is a non-convex roblem. Consequently, sub-otimal solutions of the roblem heuristic algorithms are roosed 2]. In the resent study, we solve this non-convex roblem by rewriting the non-convex objective function of the sum rate as a difference of two concave functions. The new formulation oftheroblemcanbesolvedusingaclassoftheglobalotimization methods called difference of two convex functions rogramming or shortly D.C. rogramming3]. Because of the nice roerties of the D.C. functions, D.C. rogramming attained a great attention during the last few decades several efficient algorithms for a variety of alications are roosed3],4]. Our aroach uses a branch bound algorithm. The algorithm initially estimates an uer bound a lower bound ofthed.c.functionofthesumrateoverthewholefeasible region. Then it slits the feasible region recursively into subregions where it estimates the bounds for every subregion. Considering the highest lower bound, many subregions with loweruerboundsarenotofinterest.alsoifthisbound reaches the highest uer bound for a certain artition of the feasible region with some insignificant difference, the corresonding ower allocation is taken as the otimum. Thefruitionofthealgorithmisbasedonagoodestimation ofthebounds.byintroducinganewvariable,i.e.,addinga new dimension to the roblem, the D.C. roblem is reformulated as a convex maximization roblem over a iecewise linearly outer aroximated convex set. An uer bound of a subregion is found by comuting the greatest distance with resecttothenewaddeddimensionbetweenaointinthe convexsetcorresondingtoalocallowerboundaointin the enveloe of the convex set. Furthermore, a corner oint in a subregion which leads to the highest sum rate is considered asalowerbound. In5], the authors find a global otimum ower allocation for multiuser DSL networks. Assuming a ower constraint er user, the otimization roblem for maximizing the sum rate is decouledacrossalltonestomakeitsolvableonaertone basis.sotheyformad.c.rogramoutofadualformofa weightedsumrateforasingletonewithweightedowersof all users. These weights are udated iteratively to meet both ower rate constraints. The D.C. roblem is solved using a rismatic branch bound algorithm which is roosed in 3]. Theremainderoftheaerisorganizedasfollows.Thenext section describes the system model. Section III introduces the roblem statement the D.C. formulation. In Section IV the roosed algorithm is described. Section V resents some simulation results. The conclusions are drawn in Section VI. 367

3 Tx 1, 1 Tx K, K g 1K g K1 g 11 g KK σ 2 1 σ 2 K Rx 1 Rx K Fig. 1: An interference channel scenario containing K transmitter-receiver airs II. SYSTE ODEL A general memoryless interference channel with erfect channel knowledge at the transmitters is assumed. A scenario consisting of K transmitter-receiver airs which are couled by interfering links is considered. Let h kl be the channel coefficient of the link between the transmitter l the receiver k,where k, l = 1,..., K.Thenthecorresondingchannelgain isdenotedas g kl = h kl 2.Fig.1showsaK transmitterreceiverairsscenariowhere σk 2isthenoiseoweratreceiver k. Then g 11 g 12 g 1K g 21 g 22 g 2K G = g K1 g K2 g KK is the gain matrix where the matrix s non-diagonal elements corresondtotheinterferinglinks.let = 1,, K ) T be the vector of the transmitted owers. The main concern here is how to allocate owers to the individual transmitters under the total ower constraint k = tot, k 0 1) which directly reflects the resulting total interference. The sum rateisusedasameasureoferformancewhichiscalculated based on a simlified assumtion that the interference is treated as white Gaussian noise: g kk k C = ld 1 + σk ) g kl l III. PROBLE STATEENT Based on the assumtions stated on Section II, the otimum owerallocationvector ot isfoundbysolvingthefollowing maximization roblem: g kk k ot = argmax ld 1 + σk 2 + g kl l 3) l k l k k = tot, k 0. 4) This roblem is non-convex a closed form solution is not known. Using a quotient roerty of the logarithms which statesthat loga/b) = loga) logb)for A, B > 0,the sumratefunctioncanbewrittenasadifferenceoftwoconcave functions. Therefore, the maximization roblem of3)-4) can be reformulated as a D.C. roblem: where ot = argmax f ) g )} 5) f ) = g ) = k = tot, k 0 6) ) K ld σk 2 + g kl l, 7) l=1 ld σ k 2 + l k g kl l. 8) Both functions of7) 8) are concave functions. IV. BRANCH AND BOUND ALGORITH Inthissection,analgorithmbasedonthebranchbound technique is described. The algorithm basically finds a global otimum ower allocation by searching over a full binary tree. A. Constructing the Tree Definition 1. A K-simlex T has the K + 1 vertices v 0),, v K) with v 0) beingthevertexattheorigin thereresentation x = K k=0 λk) v k) isuniqueforall x T where k λk) = 1 0 λ k) 1for k = 0,, K. The root of the tree corresondsto an initial K 1)- dimensionalface F 1),i.e., F 1) isobtainedbysetting λ 0) = 0 in a K-simlex, which covers exactly the whole feasible regionoftheowerallocationstheverticesof F 1) are the corners of the feasible region, i.e., F 1) = tot, 0,, 0),0, tot, 0,, 0),, 0,, 0, tot )] = 1), 2),, K)]. Eachnode ihastwochildren.theyareconstructedbyslitting F i) overitslongestedge.thisrocessiscalledbranching.now,consideranode iwith F i) = 1),, K)] whichhas Kverticesthelongestedgeisinbetweenthe vertices x) y).thenthenewvertexwhichisshared by F 2i) F 2i+1) is k) = 1 2 x) + y)).thetwonew K 1)-dimensional faces are F 2i) = 1),, x), k),, K)], 10) 9) 368

4 F 1) 3 3) F 2) F 3) F 4) F 5) F 6) F 7) F 1) 2) 2 Fig. 2: The structure of the tree where every node corresonds toafacecoveringthewholeorartofthefeasibleregion. F 2i+1) = 1),, y), k),, K)]. 11) ThestructureoftheresultingtreeisshowninFig.2.For examle, in a three-user scenario, the initial 2-dimensional face is F 1) = tot, 0, 0),0, tot, 0),0, 0, tot )] = 1), 2), 3)]. Itisslitintotwofaces F 2) = F 3) = 12) 1), 3), 4)], 13) 2), 3), 4)] 14) where 4) = 1 1) + 2)). 15) 2 This rocess is illustrated in Fig. 3. B. Searching Over the Tree Aly a breath-first search through the tree. For every node i in the tree, comute an uer bound u i) of max f ) g )} a lower bound l i) of max f ) g )}withthecorresondingowerallocation LB,i.Thecomutationofanuerboundalowerbound is described in details in Sections IV-C IV-D, resectively. This rocess is usually called bounding. Then udate the global lower bound as β i) = max β i 1), l i)} where β 1) = l 1) itscorresonding owerallocationis ot,i.accordingly,therearethreecases: 1) If u i) β i) < 0theowerallocationwhichmaximizes f ) g )isnotin F i).so,thereisnoneedtoinsect the children. 2) If u i) β i) > ǫforsometolerancevalue ǫ > 0,the face F i) maycontainaowerallocationcorresonding to the globalmaximumbut a lower boundwhich is closetotheuerbound u i) withsomearbitrarysmall 1 1 1) 1) 3 a) F 1) 3) F 2) F 3) 4) b) F 2) F 3) 2) Fig. 3: An examle of slitting a 2-dimensional face. difference ǫstillhastobefound.sothetwochildren nodes 2i 2i + 1needtobeinsected. 3) If 0 u i) β i) ǫthelowerbound β i) isthe localmaximumof F i) withsomeaccetablerecision if ot,i F i) itcanbeaglobalmaximumifno other nodes with higher lower bounds are found. Finally, the algorithm terminates when no more nodes have to beinsectedwithaglobalotimumowerallocation ot,i corresondingto β I) where Ireresentstheindexofthelast checked node. C. Comuting an Uer Bound Let i) = i 1) max,i 1 }with 1) = V F 1)) reresenting a set of feasible ower allocations where max,i 1 isafeasibleowerallocationwhichisfoundwhen calculatingtheuerbound u i 1) V F i)) isthesetof verticesoftheface F i). } f i) ) = min j ) T f j ) + f j ) j i) 2 16) 369

5 asanaroximationoftheconcavefunction f )withthe following roerties: f i) )isiecewiselinearconcave. ) f )isanouteraroximation. f i) f i) ) = f ), i). From i 1) i) follows f i 1) From f i) ) f )follows: ) > f i) ). } max f ) g )} max f i) ) g ). 17) } As a result, max f i) ) g ) is an uer bound of max f ) g )}. } The maximization roblem max f i) ) g ) with the total ower constraint can be reformulated as a convex maximization roblem: t u, u ) = argmax t g )} 18) t, k = tot, k 0, 19) t f i) ) 0. 20) The two constraints in19) 20) reresent a new feasible region which is described as a olytoe } E i) =, t) : a i) t A i) b i), k = tot, 21) i.e., the region is closed has linear boundaries, where I K K f 1 ) T A i) =., 22) ) T f i) ) b i) = a i) = 0K 1 1 i) 1 0 K 1 f 1 ) 1 ) T f 1 ).. ) ) T f i) i) f i), 23) ). 24) Fig.4showsthefeasibleregion E i) ofatwouserscenario.theotimumoint, t)intheolytoe E i) which maximizes t g ) has to be at the convexenveloeof E i) where t = f i) ). Now consider the oints, t) which lead to a constant value t g ) = c i) where ϕi) ) 1), f i) 1) )) 1 u, t u ) 1) t E i) t = f i) ) 2), f i) 2) 2) )) Fig.4:Thefeasibleregion E i) oftheconvexmaximization roblem of a two-user scenario. c i) = max f i) k) ) g k))}.theseointsformakdimensionalconcavefunction ϕ i) ) = t = } c i) + g )at therangeof twith = : k = tot asshownin Fig.4.Thenthemaximumof t g )corresondstothe oint u, t u )ontheenveloeof E i) whichhasthegreatest distanceto ϕ i) )withresecttot-axis. A K-dimensionalsubsace Υ i) isuniquelydefinedbythe oints k), t k)) theoriginas 2 t h T = 0, 25) where hisak-dimensionalvectortheoints k), t k)) arefoundas t k) = c i) + g k)), 26) for k = 1,, K. For the sake of simlicity, consider thesubsace Υ i) insteadoftheconcavefunction ϕ i) ). Since ϕ i) )isconcave Υ i) intersects ϕ i) )atthe oints k), t k)), } every oint, t) in Υ i) with = : k = tot leadsto t g ) c i).fig.5showsthe intersectionbetweenthefeasibleregion E i) thesubsace Υ i) whichissannedbythevectors u v.itshowsthatthis intersectionisuerboundedby ϕ i) ).Ontheotherh, thegreatestdistance i) inthedirectionoft-axisbetween aoint, t)ontheenveloeof } E i) aoint, t)in Υ i) with = : k = tot is larger than the distance between ϕ i) )anyointontheenveloeof E i) intaxisdirection.therefore, i) +c i) isasuitableuerbound of t g ). Proosition1. Thegreatestdistance i) max between a oint, t)ontheenveloeof E i) aoint, t)in Υ i) with 370

6 } = : k = tot is found using the linear rogram } ) Λ i) max, t max = argmax t λ k) t k) 27) Λ i),t a i) t A i) P i) Λ i) b i), 28) 1 1 K P i) Λ i)) = tot, 29) where Λ i) = λ 1),, λ K)) T is a vector of weighting factors λ k) ofthevertices k) P i) beingamatrixwith columns k), k. Proof:Thegreatestdistance i) max is described as max,i, t max ) = argmax t h T } 30),t), t) E i). 31) Basedondefinition1,everyowerallocation F i) is uniquely reresentable as = λ k) k) 32) with K λ k) = 1, λ k) 0, k = 1,, K.Thensubstituting 32)to t h T gives K t h T λ k) k) = t λ k) h T k). 33) Since Υ i) ϕ i) )attheoints k), t k)),theseoints satisfy h T k) = t k) c i). 34) Substituting34) in33) gives t λ k) t k) c i)) = t λ k) t k) +c i) = i) +c i) 35) where λ k) = 1.Usingtheresultof35),theotimization k roblem of30)-31) is equivalent to the linear rogram of 27)-29). The linear rogram of27)-29) can be solved using the active-setmethodwithaninitialoint k), t k)) corresondingto c i) 6].Thentheuerboundis where u i) = c i) + i) max, 36) i) max = t max max,i, t max ) = λ k) max tk), 37) )) P i) Λ i) max, f P i) Λ i) max 38) 1), f i) 1), t 1)) 1 u, t u ) u 1) t max,i, t max ) i) max ϕ i) ) )) 2), f i) )) 2) 1) Υ i) E i) v 2), t 2)) 2) Fig.5:Thesubsace Υ i) withbasisvectors u vintersects the feasible region E i) in between the oints k), t k)) where this intersection is lower than the concave function ϕ i) )int-axisdirection. isthecornerointontheenveloeof E i) withthegreatest } distanceto Υ i) with = : k = tot in the directionoft-axis max,i = P i) Λ i) max is a ower allocation whichcanbealiedasalowerboundcidate. D. Comuting a Lower Bound Foreachnode i,alowerboundiscomutedas l i) = max f ) g )}, 39) the corresonding ower allocation is with LB,i = argmax f ) g )}, 40) 2 V F i)) }, max,i. 41) V. NUERICAL RESULTS In this section, the erformance of the roosed algorithm isdemonstratedasafunctionoftheseudosignaltonoise ratio γ SNR whichisdefinedastheratioofthetotaltransmit owertothenoiseoweratthereceiversindecibel γ SNR = 10log tot /σk 2 ), k. For the following, well known sub-otimal ower allocation schemes are used as benchmarks. The first scheme is the greedy ower allocation which serves only the user with the highest channel gain: k = tot k = k max, k 0 otherwise 42) 371

7 C/bit with k max = argmax g kk }, k. 43) k Also the equal ower allocation which serves all users with equal owers is considered: k = tot, k. 44) K The third scheme is the signal to interference ratio balancing schemesir balancing) which equalizes the signal to interference ratios at all receivers: γ k) = g kk k = γ 45) g kl l l k where γ is the resulting signal to interference ratio. As introduced in1], the waterfilling scheme neglects the interference art in2) solves the resulting convex roblem of3)-4) using the Lagrangian multilier method. The allocated ower at k-thuseris } k = max 0, w σ2 k 46) g kk where w isthewaterlevel.soitassignsowerstousers basedonthenoiseowertothechannelgainrationcr). Finally, an iterative algorithm which calculates the interference ower based on the ower allocation of the revious iteration isconsideredasroosedin2].atthe j-thiteration,the interference art in2) is constant, i.e., calculated from the ower allocation in iteration j 1. The otimization roblem of3)-4) is solvable using the Lagrangian multilier method. Theassignedowertothe k-thuserat j-thiterationis σ2 k + g kl j 1) l j) k = max 0, j) w l k 47) g kk where w j) isthewaterleveliniteration j.itisshownin 2] that excluding some users, i.e., allocates no ower to some users, alying this algorithm to the other users increases the sum rate significantly. Therefore, we imlement thisalgorithminsuchawaythatitallocatesowerstoasubset of users aiming at achieving the highest sum rate. Aart from the conventional distributed iterative waterfilling roosed in 7], this algorithm is also called iterative waterfilling in the sensethatitrefillsowerstousersonthetoofthenoiselus interference to the channel gain ratiosnicr), i.e., similar to waterfilling. Assuming equal noise owers for all users, a scenario of three transmitter-receiver airs is considered with G = ) Fig.6showsthesumrateachievedatdifferent γ SNR using different ower allocation schemes. Because the SIR balancing branch bound algorithm iterative waterfilling greedy allocation SIR balancing waterfilling equal ower γ SNR /db Fig.6:Sumrateversustheseudosignaltonoiseratio γ SNR for different ower allocation schemes. scheme serves all users the high interference to the first user caused by serving the second user, the achieved sum rateislowascomaredtotheotherschemeseseciallyat high γ SNR.oreover,alyingequalowerallocation,the achieved sum rate is low but increases monotonically with γ SNR tillitsaturatesathigh γ SNR. Waterfillingservesthefirstuserorthefirsttwousersatlow γ SNR,i.e.,thethirduserhasahigherNCRascomaredtothe waterlevel.butathigh γ SNR itservesalluserswithalmost equalowersbecausethencr sforallusersareverysmall as comared to the assigned owers. Therefore, waterfilling equalowerallocationarealignedathigh γ SNR. Greedy ower allocation achieves high sum rates by serving onlythefirstuseratlow γ SNR.Butatmoderateinterference, both iterative waterfilling the branch bound algorithm achievehighersumratesbyservingboththefirstuserthe third user. Finally, because the iterative waterfilling is a sub-otimal ower allocation scheme, it achieves lower sum rates as comared to the branch bound algorithm which outerforms the other schemes reaches the global otimum ower allocation. VI. CONCLUSION In this aer, a D.C.difference of two concave functions) formulation of the non-convex otimization roblem of ower allocation aiming at maximizing the sum rate with a total ower constraint is resented. A branch bound algorithm withagoodestimationoftheboundsis roosed.foran uer bound, both the convex maximization formulation of the roblem the iecewise linear aroximation of the 372

8 constraints relax the roblem into a linear rogram roblem. The results show that the roosed algorithm reaches the global otimum as well as it outerforms other subotimum schemes in all SNR values. ACKNOWLEDGENT This work is suorted by the German Research Foundation DFG. REFERENCES 1] T.CoverJ.Thomas,ElementsofInformationTheory,2nded. John Wiley&Sons,July ] H. Al-Shatri, N. Palleit, T. Weber, Transmitter ower allocation for otimizing sum caacity interference channels, in 14th International OFD-Worksho, Setember 2009, ] R. Horst, P. Pardalos, N. Thoai, Introduction to Global Otimization, 2nd ed., ser. Nonconvex Otimization its Alications. Kluwer Acadimic Publishers, January 2000, vol ] R. Horst V. Thoai, DC rogramming: Overview,, in Journal of Otimization Theory Alications, vol. 103, no. 1, October 1999, ] Y. Xu, T. Le-Ngoc, S. Panigrahi, Global concave minimization for otimal sectrum balancing in multi-user DSL networks, in IEEE Transactions on Signal Processing, vol. 56, no. 7, July 2008, ] P. Gill, W. urray,. Saunders,. Wright, Procedures for otimization roblems with a mixture of bounds general linear constraints, in AC Transactions on athematical Software, vol. 10, no. 3, Setember 1984, ] W. Yu, G. Ginis, J. Cioffi, Distributed multiuser ower control for digital subscriber lines, in IEEE Journal on Selected Areas in Communications, vol. 20, no. 5, June 2002,