Throughput Capacities and Optimal Resource Allocation in Multiaccess Fading Channels

Transcription

1 Trougput Capactes and Optmal esource Allocaton n ultaccess Fadng Cannels Hao Zou arc 7, 003 Unversty of Notre Dame Abstract oble wreless envronment would ntroduce specal penomena suc as multpat fadng wc provde more callenges tan wred networks. esource allocaton sceme can be mplemented n te mult-access wreless systems to aceve better performance. Ts report presents te man result of [] by Tse and Hanly, wc caracterze te trougput capacty regon of mult-access fadng cannels, sow tat te boundary pont of te regon can be aceved by successve decodng and obtan te greedy algortm to fnd te optmal rate and power allocaton n fadng state.. Introducton Te classcal dscrete memoryless multple access cannel wt two transmtters, as ter capacty regon (, ) satsfyng < I( X ; Y X ) < I( X ; Y X ) < I( X, X ; Y ) () wt fxed probablty transtons p ( y x, x ) and for some ndependent nput dstrbuton p ( x ) p( x ). In te case of te Gaussan multple access cannel, wc can be modeled as: Y = X X Z Te capacty regon turns to be P log( ) σ P log( ) σ P P log( ) () σ were σ s te varance of te wte Gaussan nose Z, P, P are te power constrant of X, X. Te upper bounds are aceved wen X ~ (0, N P ) and X ~ N(0, P ).

2 Te moble wreless envronment provdes more callenges to relable communcaton not found n wred networks by ntroducng te tme-varyng cannel nature suc as multpat fadng, sadowng, and pat losses. A general strategy to combat tese detrmental effects s troug te dynamc allocaton of resources based on te states of te cannels of te users. Suc resources may nclude transmtter power, allocated bandwdt, and bt rates. Instead of fndng a sceme wt respect to specfc multple-access scemes, ere te paper [] addresses te problem at a more fundamental level as: wat are te nformaton teoretcally optmal resource allocaton scemes and ter acevable performance for multple access? Te paper looks at te capacty regon of te mult-access fadng cannel wt Gaussan nose, wen bot te recever and te transmtters can track te tme-varyng cannel, consders te trougput capacty for te fadng cannel, n suc case, te cannel statstcs are assumed to be fxed, and te codeword lengt can be cosen arbtrarly long to average over te fadng of te cannel, completes caracterzatons of ts capacty regon as well as te optmal resourceallocaton scemes wc attan te ponts on te boundary of ts regon are obtaned. It s sown tat every pont on te boundary s acevable by successve decodng. Snce te number of constrants defnng te capacty regon s exponental n te number of users, n order to obtan smple solutons of resource-allocaton, a specal polymatrod structure of te capacty regon s exploted, and greedy optmzaton algortm are derved to solve te problem. Besdes, te paper [] by Knopp and Humblet s a nce reference gvng te same result on te nformaton capacty and power control n sngle cell multuser communcatons, wc deals wt te equal rate reward case of te resource allocaton problem.. ultaccess Faddng Cannel odel X H X H Y (n). Z(n) X 3 H 3 Fgure. Dscrete-tme multple-access Gaussan fadng cannel Now let us consder te dscrete-tme multple-access Gaussan cannel Y = = H X Z( n) (3)

3 were s te number of users, X (n) and H (n) are te transmtted waveform and te fadng process of te t user, respectvely, and Z (n) s wte Gaussan nose wt varance ο. We assume tat te fadng processes for all users are jontly statonary and ergodc, and te statonary dstrbuton as contnuous densty and s bounded. User s also subject to an average transmtter power constrant of P... Capacty of fxed power scenaro A. Consder frst te smple stuaton were te users locatons are fxed and te sgnal of user s attenuated by a factor of wen receved at te base staton,.e., H = for all tme n. In te case of te Gaussan multaccess cannel, ts capacty regon s P S Cg P) = { : ( log( ) for every S {,..., }} (4) σ were = (,..., ) and P = ( P,..., P ) Note tat ts regon s caracterzed by constrants, eac correspondng to a nonempty subset of users. oreover, t s known tat te capacty regon as precsely! vertces n te postve quadrant, eac acevable by a successve decodng usng one of te possble orderngs. B. We now turn to te case of nterest were te cannels are tme-varyng due to te moton of te users. Wen te recever can perfectly track te cannel but te transmtters ave no suc nformaton, te capacty regon s gven by H P S {(,..., ) : ( E H [ log( )], S {,..., }} (5) σ were H = ( H,..., H ) s a random vector avng te statonary dstrbuton of te jont fadng process. An ntutve understandng of ts result can be obtaned by vewng capactes n terms of tme averages of mutual nformaton, a rgorous proof can be found n [3]... Capacty under transmtters power control We wll now focus on te scenaro of te nterest, were all te transmtters and te recever know te current state of te cannels of every user. Tus te codewords and te decodng sceme can bot depend on te current state of te cannels. A power-control polcy PC s a mappng from te fadng state space to. Gven a jont fadng state = (,..., ) for te users, PC () can be nterpreted as te transmtter power allocated to user. For a gven power control polcy PC, consder te set of rates gven by C f ( PC) { : ( E H [ log( H PC ( ))], S {,..., }} H (6) σ S 3

4 Teorem.: Te trougput capacty regon for te mult-access fadng Gaussan cannel wen all te transmtters as well as te recever ave sde-nformaton of te current state of te cannel s gven by C(P) C f ( PC) U PC F were F s te set of all feasble power control polces satsfyng te average power constrant F { PC : [ PC( H)] P } (7) E H Te above teorem essentally says tat te mprovement n capacty due to te transmtters avng knowledge of te cannel state comes solely from te ablty to allocate powers accordng to te cannel state. Fgure. A two-user trougput capacty regon as a unon of capacty regons, eac correspondng to a feasble power control PC.Note eac of tese regons s a pentagon. Te boundary surface of C (P) s te curved part. 3. Explct Caracterzaton of te Capacty egon A. Polymatrod Structure Lemma 3.4: Consder a dscrete memoryless multaccess cannel wt transton matrx p ( y x,..., x ), for any ndependent dstrbuton p ( x )... p( x ) on te nputs, te polyedron { : ( I[ Y; X( ( S )] S E} s a polymatrod. X Corollary 3.5: Te capacty regon C g P) of a memoryless Gaussan multaccess cannel s a polymatrod. Lemma 3.6: Let PC be any power control polcy. Ten C f (PC) defned n (6) s a polymatrod. Defnton 3.7: A rate allocaton polcy s a mappng from te set of jont fadng states to ; for eac fadng state, () can be nterpreted as te rate allocated to user wle te users are n state. 4

5 Lemma 3.8: For any power control polcy PC C ( PC) = { [ ( H)] : s a rate allocaton polcy s. t. ( ) C PC( ))} f E H Furtermore, for any permutaton π on E, v ( π ) = E [ v H H ( π )] v (π ) s te vertex of C f (PC) correspondng to te permutaton π, and, v (π ) s te vertex of C g PC( )) correspondng to permutaton π for eac state,. Te above structural result sows tat te regon C f (PC) can be wrtten as a wegted sum of te capacty regons of parallel tme-nvarant Gaussan cannels C g PC( )). B. A Lagrangan Caracterzaton of te Capacty egon Defnton 3.9: Te boundary surface of C (P) s te set of tose rates suc tat no component can be ncreased wt te oter components remanng fxed, wle remanng n C (P). Lemma 3.0: Te boundary surface of C (P) s te closure of all ponts soluton to te optmzaton problem max u subject to C( P) g suc tat (9) (8) s a for some postve u. For a gvenu, s a soluton to te above problem f and only f tere exsts a l, rate allocaton polcy ( ), and power control polcy PC ( ) suc tat for every jont fadng state, ( ( ), PC( )) s a soluton to te optmzaton problem max u r l p subject to r p) ( r, P) C g and E [ ( )] = H H, E H[ PC ( H)] = P, =,..., (0) were P s te constrant on te average power of user. One can nterpret u as a vector of rate rewards, prortzng te users, wc wll determne te order of te successve decodng; te vector l can be nterpreted as a set of power prces, for a gven u, l s cosen suc tat te average power constrants are satsfed. C. Optmal Power and ate Allocaton for a Fadng State Consder te problem of determnng ( ( ), PC ( )) for eac fadng state. Teorem 3.4: Consder te problem maxu x l y subject to x( g( y( ) S E ( x, y) were g s a monotoncally ncreasng concave functon. Defne te margnal utlty functons ' u ( z) u g ( z) l, =,..., u ( z) [maxu ( z)] (Here, x max(x,0) ). Ten te soluton to te above problem s gven by 0 ( x, y ) to aceve ts can be found by a greedy algortm. u ( z) dz and an optmzng pont 5

6 Fgure 3. A tree-user example llustratng te greedy power allocaton. Te x-axs represents te receved nterference level; y-axs te margnal utlty of eac user at te nterference levels. At eac nterference level, te user wo s selected to transmt s te one wt te gest margnal utlty. Here, user gets decoded after user, and user 3 gets no power at all. Te optmal receved powers for user and user are q and q, respectvely. Specalzng ts result to te case of te tme-nvarant Gaussan cannel gves exactly te proposed soluton to te optmzaton problem (0) dscussed earler. Te functon g s taken to be z u l g ( z) log( ) ten u ( z) () σ ( σ z) In terms of te receved powers q = ( p,..., p ), te optmzaton problem can be rewrtten as l max ur q subject to r ( g( q( ) S E () oreover, t sows tat te optmal soluton s aceved by successve decodng among te actual users. Any suc soluton can be represented by a permutaton π and a set of ntervals [ z, z ], =,..., of te real lne suc tat z = 0, z z s te receved power of user π (), and users are decoded n te order gven by π ( ), π ( ),..., π (). Wt probablty, te optmal power and rate allocaton s unque and s explctly gven by ( ) = dz PC ( ) = A A ( σ z) were A { z [0, ) : u ( z) > u ( z) j and u ( z) > 0} (3) j D. Boundary of te Capacty egon C(P) Lemma 3.5: Let u be a gven postve rate reward vector. Ten tere s a unque on te boundary wc maxmzes u, and tere s a unque Lagrangan power prce l suc tat te optmal power allocaton solvng (0) satsfes te average power constrants. 6

7 For any suc postve u, te above lemma mples tat we can defne a parameterzaton (u) wc s te unque rate vector on te boundary wc maxmzes u. Its value can be obtaned usng te greedy rate and power allocaton soluton, wt l cosen suc tat te average power constrants are satsfed. For te gven u and l, let u, l) and PC u, l) be te optmal soluton to te problem (0). Tus we ave Teorem 3.6: Assume tat te fadng processes of users are ndependent of eac oter. Te boundary of C (P) s te closure of te parametrcally defned surface were for { ( u) : u, u = } =,..., lk ( σ z) ( u ) = { l ( z) Fk f d dz 0 σ ( ) ( ) } (4) ( σ z) u k l ( σ z) ( uk u ) were te vector l s te unque soluton of te equatons lk ( σ z) { l z Fk f d dz = P ( σ ) ( ) ( ) } (5) 0 l ( σ z) ( u u ) u k =,...,. oreover, every pont can be attaned by successve decodng. E. Cases for trougput capacty and power allocaton n fadng cannels ) Sngle-User Cannel: It s te smple case of Teorem 3.6 wt = : = { ( z) f ( ) d} dz 0 ( σ z) λ σ = log( σ 0 k u u σ ( ) l ) f ( ) d by reversng te order of ntegraton. Usng (5), te constant u l s sown to satsfy te power constrant u σ ( ) f ( ) d = P 0 l Ts s just te classc water-fllng soluton to te problem of power allocaton over a set of parallel sngle-user cannels, one for eac fadng level. Te strategy as te caracterstc tat more power s used wen te cannel s good and lttle or even no power wen t s bad. ) axmum Sum-ate Pont: If we set u =... = u =, we get te pont on te boundary of te capacty regon tat maxmzes te sum of te rates of te ndvdual users. For ts case, te margnal utlty functons u (z) s are gven by l u ( z) = ( σ z) 7

8 We note tat for a gven fadng state, te margnal utlty functon of te user wt te λ smallest domnates all te oters for all z. Ts means tat n te optmal strategy, at most one user s allowed to transmt at any gven fadng state. Te optmal power control strategy PC can be readly calculated to be σ l ( ), f > j for all j PC l) = l l j 0, else. Te optmal rates are gven by σ lk = log( ( ) ) Fk ( ) f ( ) d, =,..., 0 σ λ k l were te constants λ s satsfy σ lk ( ) Fk ( ) f ( ) d = P, =,..., 0 λ k l Ts soluton was just te same as wc by Knopp and Humblet []. Note tat ts power control gves rse to a tme-dvson multple-access strategy. 3) ultple Classes of Users: Wle te above strategy maxmzes te total trougput of te system, t can be unfar f te fadng processes of te users ave very dfferent statstcs. Generally, one way of remedyng ts stuaton s to assgn unequal rate rewards to users. Let us consder an example were tere are two classes of users. Users n te same class ave te same fadng statstcs and power constrants; te class can represent users at te cell boundary, wle te class conssts of users close to te base staton. To mantan equal rates for everyone, we can assgn rate rewards u to all users n class (far), and u to users n class (near), wt u > u. By symmetry, te power prces of users n te same class are te same. We observe tat at any fadng state, te margnal utlty functon of te user wt te best cannel wtn eac class domnates tose of oter users n te same class. Tus te optmal strategy as te form tat at eac fadng state, only te strongest user n eac class transmts, and te two users are decoded by successve cancellaton, wt te nearby user (class) decoded frst. Ts gves an advantage to te user far away. Adjustng te rate rewards can be tougt of as a way to mantan farer allocaton of resources to te users. eferences [] D. Tse and S. Hanly, ultaccess Fadng Cannels Part I: Polymatrod Structure, Optmal esource Allocaton and Trougput Capactes IEEE Trans. Inform. Teory, vol. 44, NO. 7, Nov

9 []. Knopp and P. A. Humblet, Informaton capacty and power control n sngle-cell multuser communcatons, IEEE Internatonal Conference on Communcatons, Seattle, WA, June 995. [3] S. Sama and A. D. Wyner, Informaton teoretc consderatons for symmetrc, cellular, multple-access fadng cannels Part I, IEEE Trans. Inform. Teory, vol. 43, pp , Nov [4] T.. Cover and J.A. Tomas, Elements of Informaton Teroy, Wley Interscence, 99 [5] Steuard Jensen, An Introducton to Lagrange ultplers ttp://ome.uccago.edu/~sbjensen/ndex.tml [6] Lfang L, A.J. Goldsmt, Capacty and Optmal esource Allocaton for Fadng Broadcast Cannels Part I: Ergodc Capacty, IEEE Trans. Inform. Teory, vol. 47, NO. 3, ar. 00 9