Uplink capacity of self-organizing clustered orthogonal CDMA networks in flat fading channels

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1 Auhor manuscrip, published in "ITW-29, Ialy 29)" Uplink capaciy of self-organizing clusered orhogonal CDMA neworks in fla fading channels Romain Couille ST-Ericsson, Supelec 55 Roue des Lucioles 656 Sophia Anipolis, France Mérouane Debbah Alcael-Lucen Chair, Supelec Plaeau de Moulon, 3 rue Jolio-Curie 992 Gif sur Yvee, France merouane.debbah@supelec.fr hal-4453, version - 3 Jan 2 Absrac In his paper, we derive a deerminisic equivalen of he Shannon-ransform of cerain ype of large uniary random marices. This approximaion is exploied o evaluae he uplink channel capaciy of clusered orhogonal CDMA nework. When non-uniform power allocaion among he users of each cluser is allowed, we derive an explici ieraive waerfilling algorihm which, upon convergence, achieves he muliuser decoding capaciy. In paricular, we show ha, in a selforganizing clusered orhogonal CDMA nework, each cluser can opimize is power allocaion policy independenly of he oher clusers a he expense of a small feedback overhead. Simulaions corroborae he heoreical derivaions. I. INTRODUCTION Previous releases of he universal mobile elecommunicaions sysem UMTS) aimed o ensure qualiy of service for all users in he cells. The recen developmens of 3G proocols however push more and more owards he maximizaion of he user sum raes in he cell. 3G sysems are based on code division muliple access CDMA) which use random i.i.d. codes in he uplink and incur inerference among he users. Orhogonal codes which reduce drasically he inerference) are unforunaely no used as i would require some imporan overhead signaling o synchronize he users. One way of dealing wih his issue, which is par of he recen cogniive radio incenive [2], proposes o allow users o self-organize in small clusers e.g. here CDMA-based clusers) or coaliions [], [2] o opimize heir uplink communicaions o a cenral base saion [3], [4]. Due o he very shor disance beween he users wihin he cluser, non-inerfering ransmissions can be performed hrough he use of orhogonal codes per cluser. Those neworks are referred o as self-organizing neworks, which we shall consider in he following. Among he pas conribuions in he muli-cell CDMA case, a performance sudy of muli-cell orhogonal CDMA neworks in frequency selecive channels was carried ou in [3]. The auhors sugges ha frequency seleciviy has a sronger negaive impac on CDMA performance han he pah loss; herefore muliple small CDMA cells migh urn ou more advanageous han a large single CDMA cell. The same conclusions were given in a similar uplink random CDMA sudy in [4]. In [5], he performance of single-cell CDMA in fla-fading channels wih random codes was analyzed for differen ypes of decoders when power opimizaion is performed a he ransmier. Mos of hese sudies derive resuls under he assumpion ha he number of users per cell is large, since simpler derivaions are possible using he heory of large random marices [6]. In his paper we derive resuls for he uplink capaciy of a clusered CDMA self-organizing nework when he CDMA code marices are large and isomeric. Those resuls are based on ools from random marix heory [6] and free probabiliy [7], [8]. We will in paricular derive wo heorems which relae respecively he η-ransform and he Shannon-ransform see definiions hereafer) of such a channel model o deerminisic approximaions independen of he code marix realizaion. Anoher deerminisic equivalen of he η-ransform has been considered recenly in he frequency selecive case by Peacock e al. [] using a differen approach, referred o as incremenal marix expansions. However, he form of heir deerminisic approximaions does no allow for a rivial calculus of he Shannon-ransform, which is provided here. Up o he auhors undersanding, boh our and Peacock s resuls are differen, hough hey are shown o end o he same asympoic expression. Our resul on he Shannon-ransform allows us firs o derive explici expressions for he uplink clusered orhogonal CDMA capaciy and more imporanly o produce an ieraive waer-filling algorihm for he cell users o achieve nework capaciy under per-cluser sum power consrain where he power wihin a cluser is seen as a long erm exchangeable uiliy, see [2] for more jusificaions). We will also propose an algorihm for each cluser o auomaically reconfigure heir power allocaions, independenly of he oher clusers, when a user connecs or disconnecs o he cell. The remainder of his paper unfolds as follows: in Secion II, he fundamenal mahemaical resuls of his paper are provided. In Secion III, he clusered CDMA self-organizing nework model is inroduced. In Secion IV, he capaciy achieving power allocaion algorihm is presened. In Secion V, numerical simulaions are carried ou. Finally, in Secion VI, we draw our conclusions. Noaion: Capial boldface characers denoe marices I N by isomeric, we mean a non-necessarily square marix wih orhogonal columns.

2 hal-4453, version - 3 Jan 2 is he N N ideniy marix). Hermiian ranspose is denoed ) H. The operaor dex represens he deerminan of he marix X. The funcion δx) is he Kronecker Dirac funcion such ha δx) = if x = and equals oherwise. The firs derivaive of he funcion f is denoed f. II. MATHEMATICAL PRELIMINARIES Our main resuls rely on ools of random marix heory analysis, he η-ransform and he Shannon-ransform, which we inroduce in he following. Definiion η-ransform): Le µ be a probabiliy densiy funcion. The η-ransform ηx) of µ is defined as ηx) = µd) ) + x Noe ha he η-ransform is linked o he beer known Sieljes ransform Sx) via he relaion S /x) = xηx). Theorem : Le K, N be posiive inegers, {W i } i=,...,k be K independen N N Haar disribued complex random marices and {D i } i=,...,k be K diagonal N N non-negaive marices. Denoe µ i he empirical disribuion of he enries of D i. Then, for N large, K fixed, and for some x, he η-ransform ηx) of he empirical eigenvalue disribuion of B N = K W kd k Wk H saisfies approximaely ηx) = + x β k x)) 2) where he funcions β k x), k {,...,K} saisfy he K fixed-poin equaions β k x) = xηx)β k x) + xηx) µ id) 3) Proof: The proof relies mosly on he asympoic freeness [6] of he marices W k D k Wk H when N grows o infiniy. In his case, he R-ransform Rx) of B N is asympoically equal o he sum of he R-ransform R k x) of W k D k Wk H. This R- ransform is defined wih respec o he η-ransform as 2 R xη) = x η ) 4) η R + ) = xr + 5) x Denoe η k and R k he η- and R-ransform of W k D k Wk H. Then, from ) and 5), we have xr k + = µ k d) R k + x which leads o R k x) = x R k x) + x µ kd) 6) and, in paricular, defining β k x) = R k xη), we have β k x) = xηβ k + xη µ kd) 7) 2 for readabiliy, we will use he shorcu noaion f = fx) when no confusion is possible. which is exacly 3). Now, since R = K R k, using 4), his gives ) ηx) = + x R k xη)) = + x β k which complees he proof. The η-ransform is of no direc pracical use in our curren analysis hough i migh be essenial in he analysis of MMSE decoders for CDMA downlink, see e.g. []). To esablish he capaciy of uplink ransmissions in CDMA neworks, we need a deerminisic approximaion of he so-called Shannonransform, defined as follows. Definiion 2 Shannon-ransform): Le µ be some probabiliy densiy funcion. We define he Shannon-ransform Vx) of µ as Vx) = log + x)µd) 9) The Shannon-ransform is inimaely linked o he capaciy of a muli-dimensional channel whose marix empirical eigenvalue densiy funcion is µ. Noe also ha he Shannon-ransform is an inegral of /x ηx)). From his observaion, equivalenly o Theorem, we derive in he following heorem a deerminisic approximaion for Vx). Theorem 2: Le B N be an N N marix as defined in he condiions of Theorem. We have he following asympoic approximaion of he Shannon-ransform Vx) of B N as ) Vx) =log + x β k x) 8) + log + xηx)[ β k x)]) µ k d) ) where β k x) and ηx) are given by 2) and 3). Proof: The Shannon-ransform saisfies [6] Vx) = x ηu))du u we herefore seek an inegral form for he η-ransform. Noe firs ha x η) = K x + x β k ) ) = β k η ) Also noe from 3) ha xηβ k = and herefore ha = xηβ k xηβ k + xη µ kd) 2) xηβ k + xη µ kd) 3) Now, for any k, he derivaive along x of C k = log xηβ k + xη)µ k d) is C k = [ ηβk xη β k xηβ k ] + [η + xη ] µ k d) 4) xηβ k + xη

3 hal-4453, version - 3 Jan 2 Fig.. λ λ 2 λ 22 λ 23 λ 3 λ 32 Self-organizing clusered CDMA nework Recalling now 3) and 3), his yields C k = ηβ k xη β k xηβ k ) +η+xη ) β k = xηβ k 5) We also have log + x β k )) = η β k + x ηβ k 6) Adding his las expression o k C k, we end up wih he desired K β kη. Verifying ha V) =, we finally obain ). A. Model III. SYSTEM MODEL Consider a nework composed of K clusers 3 and a single base saion, each cluser being composed of a maximum of N users each. The users of cluser k {,...,K} are assigned orhogonal spreading codes forming he columns of a uniary N N Haar disribued marix W k. Each cluser operaes independenly regarding he allocaion of spreading codes, and herefore he W k marices are considered independen. Depending on is posiion in cluser k, user n {,...,N} induces a pah-loss λ kn in he uplink ransmission o he base saion 4. The pah-losses in cluser k are gahered in he diagonal marix Λ k = diagλ k,...,λ kn ). Moreover we assume ha he ransmission channel is frequency fla, i.e. he propagaion channel fade is consan over he ransmission bandwidh. User n in cluser k ransmis wih power p kn, he collecion of which is gahered in he marices P k = diagp k,..., p kn ). We addiionally assume he base saion experiences a background noise disribued as CN, σ 2 ). This siuaion is depiced in Figure. B. Uplink capaciy Denoing D k = Λ k P k, he uplink per-ime sample) capaciy C reads ) Cσ 2 ) = N log de I N + σ 2 W k D k Wk H 7) 3 we assume ha he clusers are formed in a self-organized manner based on he disance, see [2] 4 when here are less han N users in a specific cluser, we can freely wrie λ kn = for he virual users. Denoing µ k ) = N N n= δ λ knp kn ), from Theorem 2, when N is large, we have approximaely 5 ) Cσ 2 ) = log + K σ 2 β k σ 2 ) + log + σ 2 ησ 2 )[ β k σ 2 )] ) µ k d) 8) wih ηx) and β k x) saisfying he implici Equaion 3). We herefore have an expression of he uplink capaciy which only depends on he pah losses λ kn, k {,..., K}, n {,..., N}. This suggess ha he choice of he spreading codes W k s has asympoically no impac on he achievable uplink sum rae. We will now seek a power allocaion sraegy which maximizes he capaciy under cluser power consrains. IV. SELF-ORGANIZED POWER ALLOCATION In he following, based on relaive fairness among clusers and on individual mobile erminal capabiliies, cluser k is allocaed a oal allowed ransmi P k ; herefore N n= p kn = P k. A. Opimal power allocaion To maximize he sysem capaciy wih respec o he KN scalars p kn, we shall firs show he following lemma, Lemma : The power allocaion policy p kn = p kn opimizing he deerminisic capaciy approximaion 8) saisfies, for all k, n, p kn α = k σ2 η βk ) + λ kn η 9) where η, β k are he respecive values of η and β k when C achieves is maximum, and α k is such ha k p kn = P k. Proof: To prove his lemma, we follow Prop. 4 in []. We need o ensure successively ha, denoing Cσ 2 ) = V p,..., p KN, β,..., β K, η), 6 i) V/ β k = V/ η =. ii) C is sricly concave in he p kn. Par ii) is in fac obvious once i) is shown. Indeed, in his case, he successive derivaives of C along p kn equal hose of V along p kn plus oher erms he derivaives of V along η and β k ) which are null. Therefore, since V is he sum of sricly concave funcions of he p kn, C is sricly concave. To show i), we simply differeniae and observe ha V η = σ 2 β k σ 2 ηβ k σ 2 + ησ 2 µ kd) = 2) k V = ησ 2 ησ 2 + β k ηβ k σ 2 + ησ 2 µ kd) = 2) boh equaliies semming from he remarks 3) and 3). 5 noe ha he capaciy is he Shannon-ransform of σ 2 and no of σ 2. 6 V can be seen as a funcion of N + )K + independen variables.

4 hal-4453, version - 3 Jan 2 From ii), C admis a unique maximum and from i), when β k and η have reached his maximum, he power allocaion of he p kn s reduces o maximizing K log + σ 2 β k )+ log + σ 2 η[ β k ] ) µ k d) 22) independenly of η and he β k s. This is, for each k, i reduces o maximizing log + σ 2 η[ β k ] ) µ k d), whose soluion is he waer-filling soluion 9). To achieve he capaciy opimizing power allocaion policy, similarly o [], we propose he following ieraive waerfilling algorihm, A iniializaion, for all k, p kn = P k N, η =, β k =. while he p kn s have no converged do for k {,...,K} do Se η, β k ) as soluion of 2), 3) for n =...,N do ) +, wih αk such ha Se p kn = α k σ2 ηβ k λ kn η n p kn = P k. end while This algorihm canno be proved o converge. However, from he proof of Lemma and following he argumens in [], upon convergence, he algorihm is proved o converge o he opimal soluion. The power allocaion algorihm allows now he base saion o indicae o all users he uplink ransmission powers, provided ha i is aware of all he λ kn s. We will show in he following ha such a cenralized approach is unnecessary o achieve opimal power allocaion; precisely, we will show ha each cluser k can independenly perform power allocaion regardless of he λ jn, j k, provided ha each cluser feedbacks a single real parameer o a neighboring cluser afer he individual opimizaion process. B. Self-organized capaciy opimizaion The main ineres of he sysem model under consideraion lies in he independence of every cluser in erms of code allocaion: each cluser can behave auonomously, wihou he need for he base saion o inervene in he code allocaion policy. Now, in order o achieve he opimal uplink capaciy hroughou he K clusers, user n in cluser k mus be allocaed a ransmi power p kn = p kn. Those p kn s can be compued explicily a he base saion or by any cenral eniy aware of all he λ kn s. However, his requires a large amoun of overhead daa o be ransferred back and forh hrough he nework; his comes a a non-negligible cos and goes agains he philosophy of self-organized neworks. I is acually possible o circumven his issue by performing successive local opimizaions in every cluser and by successively ransmiing a single parameer o he oher clusers: his parameer is he updaed evaluaion of he η- ransform of B N. More explicily, we consider he following algorihm, Capaciy Cσ 2 ) [bis/s/hz] Simulaion, uniform power allocaion Simulaion, Mone Carlo allocaion Theory, uniform power allocaion Theory, opimal power allocaion SNR [db] Fig. 2. Uplink capaciy of a K = 3-clusered orhogonal CDMA, uniform and opimal power allocaion, N = 8. For user n in cluser k, λ kn = α k [n ]/[N ]) db, α = db, α 2 = 2 db, α 3 = 3 db. A iniializaion, in cluser k, se p kn = P k N, η = and β k =. while η has no converged do for cluser k {,..., K} do Compue η as in 2). Se β k as soluion of 3), from he updaed η. for n =...,N do Se p kn = n p kn = P k. α k σ2 ηβ k λ kn η ) +, wih αk such ha Transmi he updaed value η o cluser k + wih convenion K + ). end while This algorihm only requires for he clusers o ransmi he updaed value of η a each sep. Noe ha compuing η requires only o know he updaed value of k β k. This value does no need be addiionally ransmied if we assume ha each cluser sores he las value of η and infers from i he updaed k β k. This way, when a new user regisers or a curren user disconnecs from a cluser, all he ransmied powers can be quickly updaed hroughou he nework a a low feedback cos. Moreover, as will be observed in Secion V, he convergence ime of he algorihm is in general so fas ha a single round of opimizaion over he K clusers is sufficien and herefore he ouer while loop is no needed) o achieve a saisfying uplink rae. In he nex secion, he heoreical resuls discussed so far are confroned o simulaions. V. SIMULATION AND RESULTS In his secion, we verify he heoreical formula derived in Theorem 2, he power allocaion algorihm derived from Lemma and he self-organizing successive power allocaion proposed in Secion IV-B. Throughou his secion, we

5 hal-4453, version - 3 Jan 2 Capaciy Cσ 2 ) [bis/s/hz] Uniform power allocaion Ieraive waer-filling, round Ieraive waer-filling, opimal Local power allocaion SNR [db] Fig. 3. Uplink capaciy of a K = 3-cluser orhogonal CDMA, ieraive waer-filling power allocaion, N = 8. For user n in cluser k, λ kn = α k [n ]/[N ]) db, α = db, α 2 = 2 db, α 3 = 3 db. consider a K = 3-cluser self-organizing orhogonal CDMA nework wih N = 8 users per cluser. The uplink signal from user n in cluser k experiences a pah loss λ kn = α k [n ]/[N ]) db, wih α = db, α 2 = 2 db, α 3 = 3 db; his allows o have in each cluser a linear range in db) of users wih relaive pah losses varying from db o α k ) db. In Figure 2, we presen simulaion resuls based on he averaging of Mone Carlo simulaions when uniform power allocaion is applied or when he allocaed powers are issued from random Mone Carlo rials bu saisfy he power consrains); for he laer, for each of he simulaions, we define he sysem capaciy as he maximal rae achieved over, random realizaions of vecors {p,...,p KN } ordered in such a way ha p kj < p k,j+. 7 Those are compared agains he soluions given by Theorem 2 wih uniform power allocaion and he ieraive waer-filling algorihm, respecively. We observe a perfec fi, even for his low N = 8 value, beween he simulaed and heoreical resuls in he uniform power allocaion case. As for he opimal power allocaion, we observe an almos perfec fi, which will surely be more accurae if one increases he number of random power allocaions of he Mone Carlo mehod. In Figure 3, we apply he self-organizing power allocaion algorihm under he same consrains as in Figure 3, when, or an infinie number of rounds a leas a sufficien number of rounds o ensure convergence of η) are performed). We observe surprisingly enough ha a single round is enough o already achieve he opimal capaciy limi of he complee algorihm. As a consequence, he algorihm is exremely fas a providing high performance power allocaion among he K clusers. For he sake of comparison, we also propose in Figure 3 he so-called local power allocaion policy applied 7 his allows o significanly reduce he number of required simulaions by discarding obviously wrong soluions. by each cluser as hough i were alone in he nework, which is maximized by he classical waer-filling soluion, i.e. wih powers ) + k σ2 p kn = α 23) λ kn where α k ensures ha n p kn = P k. We observe ha, while in he low SNR region, his local sraegy is reasonable, i is no longer he case in he high SNR regime. Since he cos of he ransmission of he daum η is obviously very low in his regime, he self-organizing power allocaion policy is even more ineresing. VI. CONCLUSION In his paper, we provided wo novel heorems relaing he uplink capaciy of a large clusered orhogonal frequency fla CDMA nework communicaing o a single base saion o deerminisic capaciy approximaions. Those acually very accurae approximaions only depend on he pah losses of he users uplink signals. Moreover, we provided an efficien algorihm for he clusers o perform local power allocaion in order o maximize he sysem hroughpu a he cos of a single daum exchange in he nework. Simulaion resuls show a perfec fi beween Mone Carlo and heoreical resuls even when he sysem dimension is no very large. REFERENCES [] M. J. M. Peacock, I. B. Collings and M. L. Honig, Eigenvalue disribuions of sums and producs of large random marices via incremenal marix expansions, IEEE Trans. on Informaion Theory, vol. 54, no. 5, pp. 223, 28. [2] J. 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