Uplink Macro Diversity of Limited Backhaul Cellular Network

1 Uplink Macro Di v ersity of Limited Backhaul Cellular Network Amichai Sanderovich ∗ , Oren Somekh † , H. V incent Poor † , and Shlomo Shamai (Shitz) ∗ ∗ Department of Electrical Engineering, T echnion, Haifa 32000, Israel † Department of Electrical Engineering, Princeton U ni versity , P rinceton, NJ 08544, US A Email: amichi@tx.technion.ac.il, orens@princeton.edu, poor@princeton.edu, sshlomo@ee.technion.ac.il Abstract In this work new ach iev ab le rates are derived, for the up link ch annel o f a cellular network with joint multicell proc essing, where unlike pre v ious results, the ideal backhaul network has ﬁnite capacity per-cell. Namely , the cell sites are linked t o the central joint processor via lossless links with ﬁnite capacity . The cellular network is abstracted by symmetric models, which render analytical treatment plausible. For this idealistic m odel family , achie vable rates ar e p resented for cell-sites that use co mpress-an d-forward schemes comb ined with loca l dec oding, for both G aussian and fading channels. The r ates are g iv en in closed fo rm for the cla ssical W yn er mod el an d the sof t-handover model. T hese rates are th en dem onstrated to be rath er close to the optima l unlimited bac khaul joint processing rates, alre ady for mo dest backha ul capacities, supporting the po tential gain offered by the joint multicell processing approach. Particular attention is also given to the low-SNR character ization of these rates throug h which the effect of the limited backh aul n etwork is explicitly revealed. In addition, th e r ate at which th e b ackhau l cap acity sho uld scale in order to main tain the original hig h- SNR chara cterization of an unlimited b ackhau l capacity system is found. Index T erms Distributed An tenna Array , Fading, Limited Backhaul, Multicell Processing, Multiuser Detection, Shannon Theo ry , W yn er’ s Cellular Mod el. I . I N T RO D U C T I O N The growing demand for ubiq uitous access t o high-data-rate services has produced a sig- niﬁcant amount of research analyzing the p erformance of wireless comm unications systems. Cellular s ystems are of major interest as the most common method for providing conti nuous services to mobil e users, in both indoor and outdoor en vironments . In particular , the us e of joint This work was presented in part at the IEEE 2007 ISIT , June 2007, Nice, France. This research was supported by a Marie Curie Outgoing International Fellowship and the NEWC OM++ network of excellenc e within the 6th and 7th European Community Framewo rk Programmes, respecti vely , by the U.S. Nati onal Science Founda tion under Grants ANI-03-38807 and CNS-06-25637, and the REMON consortium for wireless communication. 2 multicell processi ng (MCP) h as been identiﬁed as a key tool for enhancing sys tem performance (see [1] [2] and references therein for recent results on MCP). Analysis of M CP has b een so far based pri marily o n the assumpt ion that all the base-stations (BSs) in the network are connected to a remote central processor (RCP) via an ideal backhaul network that is reliable, has inﬁnite capacity and full connectivity . In this case, the set of BSs ef fectiv ely acts as a multiantenn a transmitt er (downlink) or recei ver (uplink) w ith the ca veat that the antennas are geographically distributed over a lar ge area. Since the assumpt ion of an ideal backhaul network is quite u nrealistic for lar ge networks, more recently , t here h a ve been attempts to alleviate some of t he conditio ns by considering alternative models. In [3] a limited-connectivity backhaul network model is studi ed in which onl y a s ubset of neighboring cells is connected to a central jo int processor . In [4][5 ] a topological constraint is im posed in which there exist u nlimited capacity l inks on ly between adjacent cells, and m essage pass ing techniques are employed i n order to p erform joint decoding in the u plink. Finally , [6] deals with practical aspects of lim ited capacity backhaul cellular systems incorporating MCP , where each BS quantizes its receiv ed s ignal and forwards it to the RCP via a ﬁnite-capacity reliable link. It is noted that [6] uses a simple q uantization s cheme th at do es not use the correlation between the receive d si gnals at neighbori ng BSs to reduce the compressio n rate. Since i ts introduction in [7], the W yner cellu lar model family has provided a frame work for many s tudies dealin g with mul ticell processing. Despite its sim plicity , t his model captu res the essential structure o f a cellul ar system and facilitates analytical treatment. The uplin k channels of the W yner linear and planar models are analyzed in [7] for optim al and linear mini mum mean sq uare error (MMSE) M CP receiv ers, and Gaussian channels. In [8], the W yner m odel is extended to include fading channels and the performance of single and two cell-sit e processin g under various situations is addressed. In [9] the results o f [7] are extended to in clude ﬂat fading channels (the reader is referred t o [1] for a comp rehensiv e survey of studies dealing with the W yner mod el family). As mentioned earlier , most w orks dealing with M CP assume that the backhaul network connecting the cell-sites t o t he RCP is error-free with in ﬁnite capacity . In thi s work we us e the new , recently presented result from [10] to relax this assum ption and allow each cell-site to connect to the RCP v ia a reliabl e error-f ree connectio n, but with lim ited capacity . Such a m odel suits cellu lar networks, where joint decoding can improve th e overall network performance, with the underlying assum ption that the recei ved signals are forwarded to one location to be jointly processed. Since network resources are ﬁnite, in particular when the cell-s ites are in fact “hot spots” with limi ted complexity , the inclusion of ﬁnite backhaul resources facilitates better prediction of the performance g ain offer ed by MCP . Recently , the common problem of nomadic terminals sending information to a remote destination via agents wit h lossless connections has been in vestigated i n [11] and in [10] and then extended in [12] and [13] t o in clude multip le-input m ultiple-outpu t (M IMO) channels. The main di ff erence between the scalar channels and the vector channels, is the in ability t o g et a tight upper bound, due to the entropy power inequality . These works focus on the n omadic regime, i n wh ich t he nomadic termi nals use codebooks that are unknown to t he agents , but are fully known to the remote desti nation. Such setting suits the uplink channel of the limited backhaul cell ular sy stem with MCP , wh ere t he ob livious cell -sites play the role of t he agents. 3 Adhering t o [10],[12], we assess t he impact of limit ed backhaul capacity on the performance of two transmis sion schemes: (a) the “obl ivious” scheme - i n which the cell -sites are unaware of the users’ codebooks , and use a distri buted W yner-Zi v compress-and-forward s cheme to send a com pressed version of their received signal to the RCP for join t decoding; and (b) the “partial lo cal decoding” scheme - in wh ich th e users s plit their messages into t wo p arts, where the ﬁrst part is decoded according to the “oblivious” scheme, whi le the second part is decoded locally by the rele vant BSs (t reated as a version of the broadcast channel). Throughout t his work we use two variants of the li near W yner cellu lar setup [7], which provide a hom ogenous framework with respect to the mobi le users and cell-s ites. The ﬁrst model is the circular W yner model where th e cell s are arranged on a ci rcle and each user “sees” three BSs, it s local BS and the the two neighboring BSs. The second m odel is referred to as the circular soft-hando ff (SH) model, in which the cells are arranged on a circle but the users “see” only two BSs, their l ocal BS and the left neighboring BS. This setu p focuses on users that are lo cated on a cell edge, and thus are si multaneously received by two BSs. This type of situation is often referred to as soft-handoff. Due to its simpl icity and the uniq ue two-block diagonal structure of its channel t ransfer matrix, the SH m odel fac ilitates analytical treatment whi ch provides additional i nsight. For both of these setups we are in terested in t he asymptotic s cenario of inﬁnitely many nodes, in which case the circular and linear models are equi valent. W e consider both non-fading (Gaussian) and ﬂat Rayleigh fading channels, although mos t of the resul ts apply alm ost verbatim to other fading dist ributions. Special attention is give n to the low-signal to noise ratio (SNR) characterization of t he resulting achiev able rates. W e apply the tool s of [14] and provide closed-form expressions for the minimum ener gy per-bit re quired for reliable communication, and to the lo w-SNR rate slope for t he suggested t ransmission s chemes. In particular , the low-S NR p arameters are expressed in a uniform way for both the l imited and unlimi ted backhaul versions. This, in tu rn, reve als the effects of the l imited backhaul network in a si mple and concise manner . The high-SNR regime is also studied. In p articular we provide the rate in which t he backhaul capacity sh ould scale wit h the SNR, in order for both schem es to preserve the high-SNR characterization of their parallel un limited backhaul capacity counterparts. The rest of the paper is or ganized as follow . In Section II we deﬁne th e system models. Section III inclu des a s hort re view of useful previous results that are used in the sequ el. In Section IV we deriv e an achie vable rate for oblivious cell-sites, while in Section V we extend the result to the case where cell-sites can perform local decoding. Numerical examples are presented i n Section VI, which demons trate the eff ect of lim ited backhaul capacity o n M CP performance. Sev eral proofs and the analysis of the low-SNR characterization, are relegated to the append ices. I I . S Y S T E M M O D E L Throughout this paper we consi der two va riants of the l inear W yner cellul ar uplink channel [7] (see also [15]): (a) the circular W yner m odel, and (b) t he “soft-hando f f” mo del. Since both models resemble one another to a certain extent, the circular W yner model i s presented in full while the “soft-hando f f” model is brieﬂy described emp hasizing its uni que characterization. See Figure 1 for a simple illust ration of the network t opology . 4 A. T he Cir cular W yner Model This fully synchronized m odel in cludes N identical cells , indexed by j = 0 , . . . , N − 1 arranged on a circle. Each cell includes K identical single antenna mobi le users, in dexe d k = 1 , . . . , K , and a si ngle-antenna base-station . According to the wi de-band (WB) transmissi on scheme used, all transm itters sim ultaneously use all bandwidth. It is n oted t hat the WB access protocol is optimal [8]. The us ers of the j th cell transmit { X j,k } K k =1 into the chann el. Each user sends the mess age M j,k to the RCP , where M j,k ∈ [1 , . . . , 2 nR j,k ] and R j,k is deﬁned as t he communication rate. Th e rate region is deﬁned to be achiev able, if the probabili ty of erroneous message in the RCP can be made arbitrary small, for sufﬁ ciently large block length. Each cell-sit e recei ves the faded transm ission of it s cell’ s users wit h i ndependently faded interference from the users of th e adjacent cells and independent white addi tiv e noise. The recei ved si gnal at t he j th cell-site for ti me index t reads Y j ( t ) = K X k =1 a j,k ( t ) X j ( t ) + α K X k =1  b j,k ( t ) X [ j − 1] N ,k ( t ) + c j,k ( t ) X [ j +1] N ,k ( t )  + Z j ( t ) , (2-1) where [ j ] N , j mo d N , and t he ﬁxed inter-cell interference factor is α ∈ [0 , 1] . Each addi tive noise Z j ( t ) s ample is a z ero mean circularly-sym metric com plex Gauss ian random var iable with unit variance, i.e. Z j ( t ) ∼ C N (0 , 1 ) . The users use zero mean circularly-symm etric compl ex Gaussian codebooks with average power P /K , X j,k ∼ C N (0 , P K ) . The f ading coef ﬁcients { a j,k , b j,k , c j,k } K k =1 are independent and identically distributed (i .i.d.) among different us ers and and can be viewed for each user as ergodic independent processes with respect to the time index. All the users are unaware of t heir instantaneou s fading coefﬁ cients and are not allowed to cooperate in any way . Using v ector representation, expression (2-1) can be re writt en as (with th e time i ndex dropped for the sake of bre vity) Y N = H X N K + Z N , (2-2) where N = { 0 , . . . , N − 1 } and K = { 1 , . . . , K } . Accordingly , Y N = { Y 0 , . . . , Y N − 1 } i s the N × 1 recei ved signal vector , X N K = { X 0 , 1 , X 0 , 2 . . . , X N − 1 ,K − 1 , X N − 1 ,K } is the N K × 1 transmit vector X N K ∼ C N (0 , P /K I N K ) 1 , and Z N = { Z 0 , . . . , Z N − 1 } is the N × 1 noi se vector Z N ∼ C N (0 , I N ) . The matrix H is the N × N K channel t ransfer m atrix deﬁned by H =             a 0 α c 0 0 · · · 0 α b 0 α b 1 a 1 α c 1 0 · · · 0 0 α b 2 a 2 α c 2 . . . . . . . . . 0 α b 3 . . . . . . 0 0 . . . . . . . . . a N − 2 α c N − 2 α c N − 1 0 · · · 0 α b N − 1 a N − 1             , (2-3) 1 An M × M identity matrix is denoted by I M 5 where, a j = { a j, 1 , . . . , a j,K } , b j = { b j, 1 , . . . , b j,K } and c j = { c j, 1 , . . . , c j,K } are 1 × K row vectors deno ting the fa ding coef ﬁcients, experienced by th e K us ers of the j th, [ j − 1] N th, and [ j + 1] N th cells, respecti vely , and received by the j th cell-site. The above description relates to the WB protocol where all u sers transmit simu ltaneously . For an intra-cell time-division m ultiple-access (TDMA) protocol, only one user is active per- cell, transm itting 1 / K of the time using th e total cell transmi t power P . So for TDM A protocol we set K = 1 in (2-2) and (2-3). Each cell-sit e i s connected to the RCP throug h a unidirectional loss less li nk, with bandwidth of C j bits per channel us e. The RCP , which is aware of all t he users’ fading coefﬁcients, j ointly processes the sig nals and decodes the messages sent by all the users o f the cellular sy stem, where the cod e rate in bits-per-channel-use o f the k th user of the j th cell is R j,k . B. The“Soft-Handoff” model (SH) According to a ci rcular variant of t his m odel, N cells are arranged o n a circle. Unlike the W yner model, the users are lo cated on t he cell edges and each user i s receiv ed by t he two closest cell-sites. Hence, the channel transfer matrix o f this model is given by H sh =         a 0 0 · · · 0 α b 0 α b 1 a 1 . . . . . . . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . 0 0 · · · 0 α b N − 1 a N − 1         , (2-4) where a j = { a j, 1 , . . . , a j,K } , and b j = { b j, 1 , . . . , b j,K } are 1 × K row vectors denoting the fading coefﬁcients, experienced by the signals of the K users of the j th, and [ j − 1] N th cells, respectiv ely , when recei ved by the j th cell-site. As with the W yner model α ∈ [0 , 1] represents the int er -cell interference factor . All other deﬁnitions and assumption s made for t he W yner mo del i n t he previous sections hold for t he “soft-handoff” m odel as well . I I I . P R E L I M I NA R I E S In this section we re vi ew previous results d eriv ed for the W yner and SH models with unlimited b ackhaul capac ity ( C → ∞ ), which will be useful in t he s equel. As mentioned earlier , t he central receiv er is aware of all the users’ codeboo ks and channel state in formation (CSI), and t he users are not allowed to cooperate. Accounting for the underlyin g assumpt ions, the overall channel is a Gaus sian m ultiple access channel (MA C) with K N single antenna users and a one, N distrib uted antenna receiv er . Assu ming an opti mal joint MCP the per -cell sum-rate capacity of the unl imited backhaul W yner model is given by R = 1 N E H  log 2 det  I + P K H H †  , (3-1) 6 while the resp ectiv e rate of the SH mod el is achieved by replacing H with H sh in (3-1). Due to the unique symmetric power proﬁle o f the channel transfer matrices i n volved, the rate (3-1) is known analytically only for certain special cases, which are revie wed in the following subsections 2 . Extreme SNR behavior of various rates of interest are considered throughout thi s work. In the low-SNR regime, rates are approx imated by an afﬁ ne expression w hich is characterized by two parameters: E b N 0 min is the minim al energy which is required to reliably transmit one bit; and S 0 is the slope (at E b N 0 min ) of the rate as a function of the SNR [14]. An afﬁne expression i s als o used to approximate the rates in the h igh-SNR regime. In t his case the rate is characterized by t he high-SNR power slo pe S ∞ (or multiplexing gain), and the hi gh-SNR power offset L ∞ [17]. A. Gaussi an Chan nels, No F a ding (nf) Let us start with non -fading channels (so a i,j , b i,j , c i,j are cons tantly unity) and further focus on the asym ptotic case where N → ∞ . 1) W yner Model: The per-cell su m-rate capacity su pported by the W yn er model is given by [7] R nf = Z 1 0 log 2  1 + P (1 + 2 α cos(2 π θ )) 2  dθ . (3-2) Since the entries of 1 K H H † are independent of K for a ﬁxed P and non-fading channels, this rate is achiev able (not un iquely) by both intra-cell TDM A and WB prot ocols. The low-SNR regime o f (3-2) i s characterized by [1] E b N 0 nf min = log 2 1 + 2 α 2 ; S nf 0 = 2(1 + 2 α 2 ) 2 1 + 12 α 2 + 6 α 4 . (3-3) 2) SH Model: The per-cell sum-rate of th e SH setu p is given by [18][19] (see als o [20]) R sh nf = log 2 1 + (1 + α 2 ) P + p 1 + 2(1 + α 2 ) P + (1 − α 2 ) 2 P 2 2 ! . (3-4) As wit h the W yner model, in tra-cell TDMA and WB protocols are capacity achieving pro- tocols (not un iquely) under a total cell power constraint P . The low-SNR regime of (3-4) is characterized by [18][19] E b N 0 sh − nf min = log 2 1 + α 2 ; S sh − nf 0 = 2(1 + α 2 ) 2 1 + 4 α 2 + α 4 . (3-5) 2 The reader i s referred to [16] for further details on the intricate analytical i ssues conce rning the spectrum of large random Hermitian ﬁ nite-band matrices. 7 B. Rayleigh Fl at F ading (rf) channels Introducing Rayleigh ﬂat fading channels, intra-cell TDM A is no longer opti mal and the WB protocol is known to be t he capacity achieving transmission scheme [9]. A tight (for lar ge K ) upper bound for the per-cell sum-rate of th e WB protocol is giv en by [9 ]. This rate is equal to t he rate o f a si ngle-user SISO non-fading lin k with an addi tional channel gain due to the multiple cell-sites, which is 1 + 2 α 2 for the W yner model and 1 + α 2 for the soft handover model. 1) W yner Model: For the W yner m odel, the upper bo und for the per-cell sum -rate of the WB protocol is given by R rf − lk = log 2  1 + (1 + 2 α 2 ) P  . (3-6) On the ot her hand, the low-SNR exact characterization of the rate i s give n by E b N 0 rf min = log 2 1 + 2 α 2 ; S rf 0 = 2 1 + 1 K . (3-7) It is noted that upp er and lo wer moment bounds on the intra-cell TDM A protocol per-cell sum-rate are reported in [9] for fading channels as well. Since, these bounds are in volv ed and are ti ght only in the low-S NR region, which is already covered by (3-7), we will not use them in the s equel. 2) Soft-Handoff Model: T urning to Rayleigh ﬂat fading channels, the per-cell rate of t he intra-cell TDMA scheme is derived i n [18][19] for the special case of α = 1 (based on a remarkable result of [21] calculat ing the capacity of an equiv alent two tap ti me varying inter- symbol interference (ISI) channel) R sh tdma − rf = R ∞ 1 (log( x )) 2 e − x P dx Ei  1 P  P log 2 , (3-8) where Ei( x ) = R ∞ x exp( − t ) t dt is the exponential integral function . For th e WB p rotocol we have that the p er -cell sum -rate is tig htly upper bound ed (with increasing K ) by R sh ub − rf = log 2   1 + (1 + α 2 ) P + q (1 + (1 + α 2 ) P ) 2 − 4 α 2 P 2 /K 2   . (3-9) This result was proved in [21] for K = 1 (int ra-cell TDMA protocol) and was extended for arbitrary K in [22]. T aking K → ∞ , this rate becomes R sh rf − lk = log 2  1 + (1 + α 2 ) P  . (3-10) Finally , the low-SNR regime of the rate achieved by th e WB protocol in the SH mod el is characterized in [18][19] by E b N 0 sh − rf min = log 2 1 + α 2 ; S sh − rf 0 = 2 1 + 1 K . (3-11) 8 C. Upper Bo und The following “cut-set-like” bo und applies to b oth setups Pr oposition II I.1 F or th e two multicell setups at hand with equal limited backhaul C , th e per-c ell s um-rate is upper bounded b y R ub = min { C , R } , (3-12) wher e R is the rate supported by the r espective unlimited setup with MCP . Pr oof: This result follows by consi dering a cut-set bound [23] for two cuts, th e ﬁrst is by separating the central processor from the BSs, while the second i s by s eparating the BSs from t he M Ss. For the second cut , i t is easily veriﬁed t hat the normalized mutual information is equal to the per -cell s um-rate of the respective unli mited setup. W e refer to t his bound as a “cut-set-like” bound since we also account for the ass umption of no MSs coop eration in the MS-BS cut It is emph asized that the cut-set upper bound i s general and particular bounds for th e two setups under var ious condi tions of in terest, are achiev ed by replacing R in (3-12) with the respectiv e rates, reported in Section III. Furtherm ore, it is easily veriﬁed th at replacing R with an upper bound (such as (3-9)), results in a valid up per bound for the per-cell sum-rate. I V . O B L I V I O U S C E L L - S I T E S In this section we consider cell-sites that are obl ivious to the us ers’ codeboo ks and cannot perform local decoding. Instead, each cell-si te forwards a compressed version of Y j , namely U j , to the RCP , through the lossless link of band width C j . The RCP then receives the compressed { U j } and decodes the messages sent by all the users. A. Gaussi an Chan nels Using si milar argumentation as i n [7], it is easy to verify th at an intra-cell TDM A protocol is optimal in term s of the achiev able throughput, for the non-fading hom ogenous model considered. W e begin by stating the fol lowing achiev able rate-region for th e MA C. Pr oposition IV .1 An achiev abl e rate region for a general N user MA C wi th oblivious N cell - sites, connected to the RCP by error- free lim ited capacity link s having capacities { C j } i s given by ∀L ⊆ { 0 , . . . , N − 1 } : X t ∈L R t ≤ min S ⊆N ( X j ∈S [ C j − r j ] + I ( X L ; U S C | X L C ) ) , (4-1) where P X N ,U N ,Y N ( x N , u N , y N ) = N Y j =1 P X j ( x j ) N Y j =1 P Y j | X N ( y j | x N ) N Y j =1 P U j | Y j ( u j | y j ) , (4-2) 9 and r j = I ( Y j ; U j | X N ) . An outline of t he p roof, based on [10], appears in part I of the Appendi x and is given for any channel matrix H , i ncluding a random ergodic channel. When the channel is not a fading channel, s uch as i n Proposi tion IV .1, the proof from Append ix I is applied by taking H to be a known constant (which is als o ergodic stationary process). For the Gaussi an channel we us e { X j , U j } that are complex Gaussian and also the jo int probability (4-2) i s Gaussian. It is no ted that the Gaussian statist ics are used due to the simplicit y and rele vanc y of the reported results, with no claim of optimali ty . In fact, a b etter signalling approach is already sug gested in [10], with direct im plications here. For the Gaussian channel, the m utual informatio n i ncluded in (4-1) reduces to [10][12] I ( X L ; U S C | X L C ) = log 2 det( I + P diag(1 − 2 − r j ) j ∈S C H S C L H ∗ S C L ) , (4-3) where H S C L is the transfer matrix b etween the o utput vector Y S C and the inp ut vector X L , and r j are positive parameters that are subj ected to optimizatio n over 0 ≤ r j ≤ C j . Focusing on the setup at hand, where H i,j is zero for N − 1 > | i − j | > 1 for both W yner and SH models, equation (4-1) becomes X t ∈L R t ≤ min S ⊆ [ L +1] N ∪ [ L− 1] N ∪L X j ∈S [ C j − r j ] + I ( X L ; U S C | X L C ) , where [ L ± 1] N , { j : j = ( i ± 1) mod N , i ∈ L} . Let us deﬁne H S = H S N , which is the transfer matrix betw een X N and Y S . Hereafter , we l imit ou r attentio n to the sym metric case of C i = C for all cell-sites, and R t = R for all users. By s ymmetry and concavity , this limit s the optimal r j to be inv ariant with respect to j : r j = r , and the sum-rate inequali ty ( L = { 0 , . . . , N − 1 } ) to be the d ominant inequality in (4-1). Consequently we get the following. Corollary I V .2 An achiev able rate for bo th W yn er and SH mod els wit h equal capacity link s C , equal rate us ers and oblivious cell -sites is giv en by R obl = 1 N max 0 ≤ r ( min S ⊆N ( |S | [ C − r ] + log 2 det  I + P (1 − 2 − r ) H S C H ∗ S C  )) . (4-4) This rate is achiev ed b y compl ex Gaussian { U j , X j } . Next, we need t o calculate the l ogarithm of the determinant in (4-4). In t he case where no inter- cell interference i s present ( α = 0 ), it is easily veriﬁed that H S C H ∗ S C is an |S C | identity matrix, and in this case the rate equals t he rate achie ved by an equiva lent si ngle-user single-agent Gaussian channel [10], which is given by R obl − g = log 2  1 + P 1 − 2 − C 1 + P 2 − C  . (4-5) 10 For α > 0 , we focus on t he case where the number of cells N is lar ge. An achiev able rate for thi s asympt otic scenario is given by th e foll owing propos ition, which i s one of the central results in thi s paper . Pr oposition IV .3 An achievable rate for t he cir cul ar models with equal limited capacit ies, oblivious cell-sites and an inﬁn ite number of cells ( N → ∞ ), is given by R obl = F ( r ∗ ) , (4-6) wher e r ∗ is the s olution of F ( r ∗ ) = C − r ∗ , (4-7) and F ( r ) , lim N →∞ 1 N log 2 det  I + (1 − 2 − r ) P H H †  . (4-8) Notice t hat when C → ∞ , t hen also r ∗ → ∞ , and (4-6) reduces to the p er -cell sum-rate capacity with opti mal jo int processing and unlim ited backhaul capacity [7]. For ﬁnite C , th e implicit equati on (4-7) is easily solved n umerically , since F ( r ) i s monotonic for the symmetric models at hand. The following lemm a is required for the proof of Propositio n IV .3. T his lemma is proved in part II of the Appendi x for er godic fading channels, where taki ng H to be a known constant is a special case. Lemma IV .4 Any subs et S su ch that |S | = f ( N ) ( f : R + 7→ R + , lim N →∞ f ( N ) N = λ , 0 ≤ λ ≤ 1 ), which mi nimizes equat ion (4-4), when N → ∞ , i ncludes o nly consecutive indices (considering also modu lo operation). Denote a subset which contains only consecutive indices by S ( c ) . Pr oof of Pr oposition IV .3 (ou tline): First, note t hat by applying Szeg ¨ o’ s theorem [7], on log 2 det( I + P (1 − 2 − r ) H S ( c ) H ∗ S ( c ) ) when |S ( c ) | → ∞ , we get the following sim ple explicit expression lim |S ( c ) |→∞ 1 |S ( c ) | log 2 det( I + P (1 − 2 − r ) H S ( c ) H ∗ S ( c ) ) = F ( r ) . Let us deﬁne s = |S ( c ) | , so that log 2 det( I + P (1 − 2 − r ) H S ( c ) H ∗ S ( c ) ) = sF ( r ) + ǫ ( s ) , (4-9) where lim s →∞ ǫ ( s ) /s = 0 . Secondly , from Lemm a IV .4, wh en N → ∞ , a minimum for equation (4-4) is wi thin the subspace of s ubsets that contain onl y consecutive indices { S ( c ) } . Combining (4-9), when N → ∞ , equati on (4-4) becomes R obl = lim N →∞ ( max 0 ≤ r ( min 0 ≤ s ≤ N  N − s N [ C − r ] + s N F ( r ) + ǫ ( s ) N  )) = max 0 ≤ r  min 0 ≤ λ ≤ 1 { (1 − λ ) [ C − r ] + λF ( r ) }  . (4-10) 11 Since F ( r ) is mono tonically increasing, (4-10) is maximized by r ∗ , whi ch is deﬁned by F ( r ∗ ) = C − r ∗ . 1) Low-SNR Characterization: Next we study the low-SNR characterization of the obl ivious schemes. The analysis is general and the results are used for various channels of interest. It is noted that throug hout thi s s ection we assume that the ﬁnite backhaul capacity C is m uch lar ger than the resulting rates. Focusing on the low-S NR regime, where P ≪ 1 , the per -cell sum-rate of Proposi tion IV .3 in [bits/sec/Hz] is well approxim ated by the ﬁrst three terms of its T aylor series: F ( r ∗ ) ≈ ˙ F P (1 − 2 − r ∗ ) log 2 e + 1 2 ¨ F P 2 (1 − 2 − r ∗ ) 2 log 2 e + o ( P 2 ) , (4-11) where ˙ F , dF ( ∞ ) dP    P =0 and ¨ F , d 2 F ( ∞ ) dP 2    P =0 , are th e ﬁrst and second deriv ativ e of the unlimited backhaul rate function in [nats/sec/Hz] (when r ∗ = ∞ ) wi th respect to th e SNR P at P = 0 . Substitutin g (4-11) i n (4-7) we get the following equation: ˙ F P (1 − 2 − r ∗ ) + 1 2 ¨ F P 2 (1 − 2 − r ∗ ) 2 = ( C − r ∗ ) log e 2 . (4-12) Observing (4-7) it is clear that for low-SNR , F ( r ∗ ) is small, which means t hat C − r ∗ ≪ 1 . Hence, 2 C − r ∗ is well approxim ated by 2 C − r ∗ ≈ 1 + ( C − r ∗ ) log e 2 + o  ( C − r ∗ ) 2  . (4-13) Substitutin g (4-11) into (4-12 ) and s ome additional alg ebra we get the follo wing quadratic equation for t he rate (in nats/dimensio n) of Proposition IV .3 in the low-SNR regime 1 2 ¨ F P 2 2 − 2 C R 2 −  ˙ F P 2 − C + ¨ F P 2 (1 − 2 − C )2 − C + 1  R + ˙ F P (1 − 2 − C )+ 1 2 ¨ F P 2 (1 − 2 − C ) 2 = 0 . (4-14) Neglecting the R 2 term we have t hat R ≈ ˙ F P (1 − 2 − C ) + 1 2 ¨ F P 2 (1 − 2 − C ) 2 ˙ F P 2 − C + ¨ F P 2 (1 − 2 − C )2 − C + 1 . (4-15) Finally , by applying the deﬁniti ons of the low-SNR parameters of [14] to (4-15) and some additional algebra, we get the following propositi on. Pr oposition IV .5 The low-SNR characterization o f the channel wit h the ob livious scheme and limited backhaul capacity C is given by E b N 0 min = f E b N 0 min 1 1 − 2 − C ; S 0 = e S 0 1 1 + e S 0 2 − C 1 − 2 − C , (4-16) wher e f E b N 0 min and e S 0 ar e the minimum transmitted ener gy per bit r equired for re liable commu- nication, and the low-SNR slope of the unlimited channel F ( ∞ ) , r espectively . It is easily veriﬁed that with increasing backhaul capacity , the low-SNR characterization o f the limited channel (4-16 ) coincides wi th that of the unlimited channel. Examin ing (4-16), it can 12 also be veriﬁed that by allocating at least C ≈ 3 . 2 [bits/s ec/Hz] to the backhaul network, the minimum energy required for reliable comm unication of th e lim ited channel will not increase by more t han 0 . 5 [dB] when compared to that of the unli mited backhaul. Proposition IV .5 is especially useful in cases where (4-7) can no t be solved explicitly . Ne vertheless, we can use the few cases where (4-7) can be explicitly solved (e.g. the single- antenna single-agent Gaussi an channel (4-5)) in order to validate the result of (4-16). Indeed, extracting the low-SNR characterization of the latter c an be achie ved by applying the deﬁ nitions of [14] directly to (4-5): E b N 0 min = log e 2 1 − 2 − C ; S 0 = 2 1 − 2 − C 1 + 2 − C . (4-17) The same result can b e als o derive d by substitut ing the low-SNR characterization o f the single- user singl e-antenna Gaussian case, E b N 0 min = log e 2 ; S 0 = 2 , (4-18) into (4-16). 2) High-SNR characterization: Here we study the high-SNR characterization o f the obliv- ious s cheme. Sim ilar to the previous subs ection, the high-SNR analysis is general and the results are applicable for various channels of in terest. Speciﬁcally , the case of a fading channel is covered. From (4-7) it is evident t hat for a ﬁxed backhaul capacity C and increasing SNR P , t he rate conv erges to C . Hence, for ﬁxed C the rate i s ﬁnite and the system loses its multiplexing gain (i.e. S ∞ = 0 ). The latter can b e also concluded im mediately from the upper bound (3-12). Thus in the sequel, we are interested in the rate at which the backhaul capacity C should scale with P , in order for th e system to maintain it original (unlim ited backhaul capacity setup) hi gh-SNR characterization. Follo wing [17], the achiev able rate of the unrestricted-backhaul system ( F ( ∞ ) ) can be well approximated by th e following afﬁne expression for high SNR: F ( ∞ ) ∼ = S ∞ (log 2 P − L ∞ ) , (4-19) where S ∞ and L ∞ are the hi gh-SNR parameters from [17 ]. Observing equation (4-3), for high SNR purposes, we can make the following argument: When P ′ , P (1 − 2 − r ⋆ ) is very large, an unlim ited sy stem with P ′ has the same hi gh SNR characteristics as a lim ited-backhaul system w ith P . So that for very high P (1 − 2 − r ⋆ ) we hav e F ( r ⋆ ) ∼ = F ( ∞ ) | P ′ = P (1 − 2 − r ⋆ ) ∼ = S ∞ (log 2 P − L ∞ + log 2 (1 − 2 − r ⋆ )) . (4-20) Additionall y , since we want t he rate of the backhaul-limit ed sy stem (4-7) to scale th e same way as (4-19), we require the following asympto tic equ iv alences to ho ld: R obl = F ( r ⋆ ) ∼ = S ∞ (log 2 P − L ∞ ) (4-21) R obl = C − r ⋆ ∼ = S ∞ (log 2 P − L ∞ ) . (4-22) 13 T akin g r ∗ → ∞ as P → ∞ , such t hat P (1 − 2 − r ∗ ) → P , the right hand side of the asymptoti c equiv alence (4-20) can replace the left h and s ide of the asympto tic equ iv alence of (4-21). On the other hand, (4-22) requires t hat C ∼ = S ∞ (log 2 P − L ∞ ) + r ∗ . (4-23) Thus taking C ∼ = S ∞ (log 2 P − L ∞ ) + Θ( P ) , where Θ( P ) → ∞ as P → ∞ sufﬁces to achie ve the high-SNR characterization of t he unrestricted-backhaul network. The exact scaling of r ∗ can be very slow , for e xample log 2 log 2 P . Nonetheless, the lar ger t he gap between C and F becomes ( Θ( P ) i s increasing faster with P ), the faster the asy mptotic equiv alence in the high-SNR is achieved. Corollary I V .6 In or der to pr eserve the high-SNR characterization of th e origina l unlimited back haul capacit y setup ( S ∞ and L ∞ ), it is sufﬁcient fo r the ba c khaul capacity to scale with the SNR on th e order of C ( P ) = S ∞ log 2 P + Θ( P ) , wher e Θ( P ) → ∞ as P → ∞ , at arbitrary rate. In the next stage, we write closed-form expressions for F , using known result s for both the W yner and the SH model s. 3) The W yner Model - Gaussian Channels: For the W yner model, F ( r ) can be easily deri ved using (3-2), to be F nf ( r ) = Z 1 0 log 2 (1 + P (1 − 2 − r )(1 + 2 α cos 2 π θ ) 2 ) dθ . (4-24) So that for the unli mited scenario, the achiev able rate is indeed the joint cell-site capacity: R nf − obl = F nf ( ∞ ) = Z 1 0 log 2 (1 + P (1 + 2 α cos 2 π θ ) 2 ) dθ . Next, we consider the lo w-SNR regime for the W y ner model. The main result stated in Proposition IV .5 is that the lo w-SNR characterization in this case can b e expressed by the low-SNR characterization of th e same channel but with un limited backhaul. Using this result and th e low-SNR characterization of the non-fading unlim ited W yner model (3-3), we get the low-SNR characterization of the per-ce ll sum-rate of the W y ner upli nk channel with lim ited backhaul capacity: E b N 0 min = log 2 (1 + 2 α 2 )(1 − 2 − C ) ; S 0 = 2(1 + 2 α 2 ) 2 (1 − 2 − C ) 1 + 12 α 2 + 6 α 4 + (1 − 4 α 2 + 2 α 4 )2 − C . (4-25) From (4-25), we see that the deleterious effect o f limit ed backhaul is an increase in the minimum ener gy per-bit required for reliable comm unication and a decrease in the rate’ s low-SNR slop e. This effec t clearly dimini shes when C increases. 14 4) The SH Model - Ga ussian Channels: Following s imilar ar guments to those made for the W yner set up, and capit alizing on the fact that the rate o f the unlimit ed mo del is g iv en in an explicit closed-form expression [19], we have th e fol lowing closed form expression for the achiev able per-cell sum -rate of the uplink SH model w ith oblivious BSs, av erage transm it power P , and equal limi ted backhaul capacity links C : R sh − obl = log 2 1 + (1 + α 2 ) P + 2 α 2 2 − C P 2 + p 1 + 2(1 + α 2 ) P + ((1 − α 2 ) 2 + 4 α 2 2 − C ) P 2 2(1 + 2 − C P )(1 + α 2 2 − C P ) ! . (4-26) See Appendi x III for th e deriv ati on. Next we consider the achiev able rate R sh − obl under seve ral asympto tic scenarios. For eith er increasing C or increasing P while the ot her is kept ﬁxed, th e rate coincides wit h t he cut-set bound ( min { R sh , C } ). Applying the deﬁnitions of [14] d irectly to (4-26) we get t hat for ﬁxed C th e low-S NR regime o f R sh − obl is characterized by E b N 0 min = log e 2 (1 + α 2 )(1 − 2 − C ) ; S 0 = 2(1 + α 2 ) 2 (1 − 2 − C ) 1 + 4 α 2 + α 4 + (1 + α 4 )2 − C . (4-27) As with the W yner model, we see that the deleterious effe cts of the l imited b ackhaul is an increase in the minim um ener gy per-bit required for reliable comm unication with a correspond- ing decrease in the rate’ s low-SNR slope. These ef fects clearly dimini sh when C increases. It is n oted t hat th e same result can be obtained by applying Propositi on IV .5 and s ubstituti ng the low-SNR parameters of the unli mited SH model (3-5) in (4-16). B. F adi ng Chann els Upon the i ntroduction of ﬂat fading, the intra-cell TDMA prot ocol is n o longer optimal e ven for the unlimi ted backhaul model, and the WB protocol is th e capacity-region-achie vi ng scheme (see [9]). Pr oposition IV .7 The per-cell achievable er god ic su m-rate of the WB pr ot ocol deployed i n the inﬁnit e cir cular model with equal limited capacities and i n the pr esence of fading is given by (4-6) wher e F ( r ∗ ) = lim N − →∞ max r i : C N × N K → R + s . t . E r i ( H N ) = r ∗ E  1 N log 2  det  I N + P K diag  1 − 2 − r i ( H )  N i =1 H N H † N  . (4-28) Pr oof: The proof for th e fading channel follows along the same l ines as th e proof for the Gaussian channel in Proposition IV .3, and is based on [13]. The main differe nces consist 15 of an add itional conditio ning on H i n all th e mutual informati on expressions in Proposition IV .1 (where the proof of Propositi on IV .1 in Appendix I already accounts for t he fading), in an additional expectation with respect t o H in (4-4), and since the auxil iary variable U d epends also on t he channel, so does r (so the expectation will be also over r i ), and with an u pdate to Lemma IV .4, such that it will be sui ted to the fading chann el (where the proof of Lemma IV .4 in Appendix II, already accounts for the fading). For the sake of compactness and usability , we us e a lower bound to F ( r ∗ ) from equati on (4-28), by considering r to be a constant r ∗ , regardless of the inst antaneous channel H . This giv es F ( r ∗ ) = lim N − →∞ E  1 N log 2  det  I N + P K  1 − 2 − r ∗  H N H † N  . (4-29) Since the channel is er godic, for a large number of users K , this bound is tight (see [13]), where already at K = 2 there are 6 recei ved sign als at each cell -site and a very sm all gap i s expected. Unfortunately , the sum-rate F ( r ∗ ) (and e ven F ( ∞ ) ) is explicitl y known or can be bounded only for a fe w special cases. In the sequel we use the result s presented in Section III-B to assess the i mpact of l imited backhaul i n these special cases. 1) W yner Model: W e st art with the case where the num ber of users K per-cell i s large while the t otal cell a verage transmit power P is ﬁxed. In this case we hav e th at F ( r ∗ ) = R rf − lk ( P (1 − 2 − r ∗ )) , (4-30) where R rf − lk is giv en i n (3-6) . Solving th e ﬁxed point equation F ( r ∗ ) = C − r ∗ for (4-30), we get an explicit expression for the rate R obl − rf − lk = log 2  1 + (1 + 2 α 2 ) P (1 − 2 − C ) 1 + (1 + 2 α 2 ) P 2 − C  . (4-31) Hence, the rate of the limited net work equals t he rate of the single us er Gaus sian channel (4-5) but with enhanced power (2 α 2 + 1) P . It is noted that replacing F ( r ∗ ) wit h an up per bound that increases wit h P and equals zero when P = 0 , provides an u pper bound to the rate. Examining (4-31) it is easily veriﬁed that the rate achiev es the cut-set bound when either C or P increases whil e the other is ﬁxed. It is further no ted that the rate (4-31) is also a tight upper bound for any arbitrary number of users per-ce ll. T urning now t o the low-SNR region wit h an arbitrarily n umber of us ers per-cell, we apply the general results of Propos ition IV .5 and substi tute the low-S NR parameters of the unli mited W yner setup (3-7) in (4-16) t o get th e fol lowing per- cell sum-rate chara cterization for the W yner Mod el with the WB protocol, in the presence of Rayleigh fading: E b N 0 min = log 2 (1 + 2 α 2 )(1 − 2 − C ) ; S 0 = 2(1 − 2 − C ) 1 + 1 K +  1 − 1 K  2 − C . (4-32) Here also, the deleterious effect of the limi ted backhaul is again m anifested in an increase in the minimum ener gy per- bit required for reliable communication and a decrease in the rate’ s low-SNR slop e. 16 T o conclude this s ection we note that the extreme results obtained for large K and low-SNR can be easil y extended to include a general fading distribution and are om itted for the s ake o f conciseness. In additio n, for the intra-cell TDMA protocol ( K = 1 ) we can use the moment bounds of [9] to provide respective lower and upper bounds to th e rate in the lim ited backhaul case. Since these bounds are ti ght only i n the low-SNR regime, whi ch is basically cove red by Proposition IV .5, they are om itted as well. 2) The S oft-Handoff Model: W e start by claimi ng that the per-cell sum -rate suppo rted by the SH model and WB protocol in the presence of fading i s giv en by (4-29) while replacing the W yner channel t ransfer matrix with the SH matrix (2-4). Here, as wi th the W yner mod el, closed form expressions for t he unlimit ed backhaul case are known only for a few li mited cases (see Section III-B.2). W e con sider ﬁrst the intra-cell TDMA protocol ( K = 1 ). Using the remarkable result of [21] which calculates th e capacity for tim e-v ariant two-tap ISI channel (3-8), we have th e following per -cell sum -rate capacity for the in ﬁnite circular SH mo del with oblivious BSs, an intra-cell TDMA protocol ( K = 1 ), Rayleigh fading channels, and α = 1 F ( r ∗ ) = R sh tdma − rf ( P (1 − 2 − r ∗ )) = R ∞ 1 (log e ( x )) 2 e − x P (1 − 2 − r ∗ ) dx Ei  1 P (1 − 2 − r ∗ )  P (1 − 2 − r ∗ ) log e 2 , (4-33) where Ei( x ) = R ∞ x exp( − t ) t dt is the exponential integral functi on. Un fortunately , we are able to calculate the rate itself onl y numerically by s olving (4-6) with (4-33). T urning to t he WB protocol where all K users are activ e simu ltaneously , we use th e upper bound of (3-9) to state the following upp er bound: R ub obl − rf = log 2   1 + P (1 + α 2 ) + 2 P 2 α 2 2 − C /K + q (1 + P (1 + α 2 )) 2 − 4 P 2 α 2 (1 − 2 − C ) /K 2 (1 + P (1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K )   . (4-34) See Appendi x IV for the deriv ation. Next we consid er the upper bound R ub obl − rf under seve ral asympto tic scenarios. For C → ∞ and ﬁxed P , R ub obl − rf coincides with th e respectiv e unl imited setup (3-9). On th e other hand, for P → ∞ and ﬁxed C , the upper bound R ub obl − rf → C , achie ving the cut-set bound. In addition, it is easily veriﬁed that C should scale li ke log 2 P for R ub obl − rf to achieve the optimal multiplexing gain of 1 . Finall y , for increasing number of users K ≫ 1 , ﬁxed to tal-cell power P and ﬁnite C , the upper bou nd R ub obl − rf reduces to R ub obl − rf → K →∞ log 2  1 + P (1 + α 2 )(1 − 2 − C ) 1 + P (1 + α 2 )2 − C  , (4-35) which equals the rate of a lim ited Gauss ian sin gle user SISO channel (see (4-5)) with enhanced power P (1 + α 2 ) . It is noted that th is resul t can be derive d directly by setting F ( r ∗ ) = R sh rf − lk ( P (1 − 2 − r ∗ )) and solv ing F ( r ∗ ) = C − r , wh ere R sh rf − lk is the asym ptotic expression (with increasing K ) giv en in (3-10). Therefore, i t is concluded that the bound R ub obl − rf is tight for K ≫ 1 . 17 T o assess the impact of limited backhaul in the low-S NR re gime for R ayleigh fading channels we apply Proposition IV .5 and substitute the low-SNR parameters of th e unlimited soft-handoff, to obtain E b N 0 min = log 2 (1 + α 2 )(1 − 2 − C ) ; S 0 = 2(1 − 2 − C ) 1 + 1 K + 2 − C  1 − 1 K  . (4-36) V . C E L L - S I T E S W I T H D E C O D I N G In order to better uti lize the b ackhaul bandwidth between the cell-si tes and the RCP , we consider using local decoding at the cell-si tes. In this case t he cell-sites shoul d be aware of the associated codebooks, and thus do no t operate in the nomadic regime [10]. It is noted that in general, decoding decreases the no ise uncertainty , t hus increasing the ef ﬁciency of backhaul usage. In thi s section we present an intuit iv e, sim ple scheme which p rovides an achiev abl e rate accounting for t his l ocal processi ng. According to th is schem e, each user employs rate s plitting and divides it s message int o two parts: one t hat is decoded at the RCP and another which is decoded at the local cell-site. In this case t he message that is intended for the RCP to decode, interferes with the local decodin g of the relev ant message at the cell-site. Let the power used for the former be β P and the l atter (1 − β ) P , where 0 ≤ β ≤ 1 . There are t wo st rategies for the cell-site to exec ute: to decode o nly its local user’ s m essage, or to decode also the interfering users’ messages, emerging from the neighboring cells (see [8] Section III.D). The locall y decoded inform ation rate is denoted by R d ( β ) . Forw arding the d ecoded inform ation through t he loss less links reduces the bandwidth av ail- able for comp ression, so the achie vable rate is R sd ( C ) ( sd stands for separate d ecoding ) given by R sd ( C ) = max β n F β ( r ∗ d ) + ˆ R d ( β ) o , (5-1) where ˆ R d ( β ) = min { R d ( β ) , C } , (5-2) r ∗ d is the s olution of F β ( r ∗ d ) = C − ˆ R d ( β ) − r ∗ d , (5-3) and using equation (3-2), F β ( r ) = R nf ( β P (1 − 2 − r )) for the W yner model, or usin g equati on (3-4) F β ( r ) = R sh nf ( β P (1 − 2 − r )) for the SH m odel. For α = 0 this scheme is o ptimal, s ince t here is no in ter- cell interference and each cell-s ite can decode m essages at the same rate as the RCP can. Note that the rate R sd ( C ) is not concave in C in general, and thus tim e-sharing may im prove the achiev able rate, w hich leads t o the foll owing proposition ( ch st ands for the conv ex- hull ). Pr oposition V .1 An achievable rate of the r ate-splitti ng scheme deployed i n the inﬁnit e cir cula r W yner model wit h limi ted equ al capacit ies C , i s given by R sdch, 1 = max λ,C 1 ,C 2 : λC 1 +(1 − λ ) C 2 ≤ C { λR sd ( C 1 ) + (1 − λ ) R sd ( C 2 ) } . (5-4) 18 In fact, n umerical calculatio ns reveal th at a good strategy is to d o tim e-sharing between the two extreme approaches: us ing decoding at the cell-sites, with no decoding at the RCP , and decoding only at the RCP (4-10), rather than si multaneously using the mixed approach of (5-1). Thus, d eﬁning t = ˆ R d (0) , the rate R sdch, 1 of (5-4) can be writt en as R dec = max r ≥ r ∗  t + ( C − t ) F ( r ) − t F ( r ) + r − t  , (5-5) where F ( r ) = R nf ( P (1 − 2 − r )) , usi ng equati on (3-2) for the W yner model and F ( r ) = R sh nf ( P (1 − 2 − r )) , using equati on (3-4) for the SH model. The value of r ∗ is calculated by (4-7). A more detailed deriva tion i s given in Appendi x V. It is expected that decoding at the cell-site wi ll be beneﬁcial when α is s mall (low inter-cell interference), or wh en C is sm all, so that decoding before transm ission saves bandwidth, whi ch otherwise would hav e been wasted on noise quant ization. 3) Low SNR Characterization: T o deriv e t he low-SNR characterization ( P ≪ 1 ) of (5-5), with general rates F ( r ) and t , we u se (4-11) and t ≈ ˙ tP log 2 e + 1 2 ¨ tP 2 log 2 e + o ( P 2 ) , (5-6) where ˙ t , dt dP   P =0 and ¨ t , d 2 t dP 2    P =0 , are the ﬁrst and second deriv ative of the rate function in [nats/sec/Hz] with respect to the SNR P at P = 0 . W e st art by noting that ( C − t ) ≈ C , and since r ∗ ≈ C and th e maximization is over r ≥ r ∗ , we have also t hat F ( r ) − t + r ≈ r . Hence, (5-5) can be re written in the low-SNR regime as R ≈ t + C max r ≥ C F ( r ) − t r . ( 5-7) By substi tuting (5-6) and (4-11) into (5-7) we get R ≈ ( ˙ tP + 1 2 ¨ tP 2 + C max r ≥ C P ( ˙ F (1 − 2 − r ) − ˙ t ) + 1 2 P 2 ( ¨ F (1 − 2 − r ) 2 − ¨ t ) r ) log e 2 . (5-8) Neglecting P 2 terms and recalling that ˙ F ≥ ˙ t , it is easily veriﬁed that the maximi zation of (5-8) is achieved at r m = max { C , ˜ r m } , (5-9) where ˜ r m is the u nique soluti on to the foll owing ﬁxed p oint equation wi th respect to r : 2 − r (1 + r log e 2) = 1 − ˙ t ˙ F , (5-10) which can be re written explicitly as 2 − r (1 + r log e 2) = 1 − f E b N 0 min E b N 0 d min , (5-11) 19 where f E b N 0 min and E b N 0 d min are the minimum transmitted ener gy per-bit of the local decoding scheme and the unlimi ted s etup, respective ly . Using r m , the opt imal tim e-sharing parameter λ o (see (V .2)) can be approximated in t he low-SNR region as λ o ≈ 1 − C r m . (5-12) Furthermore, using r m the ﬁxed equation (V .1) can b e approximated in the low-SNR region by F ( r ) = C ′ − r , (5-13) where C ′ is the effec tive backhaul capacity , which can be approxim ated in t he low-SNR region as C ′ = C − λt 1 − λ ≈ r m . (5-14) Finally , by apply ing th e deﬁniti ons of the l ow-S NR parameters of [14] to second equati on of (V .1), usi ng Propositi on IV .5, and some additional algebra, we get the fol lowing propo sition. Pr oposition V .2 The general l ow-SNR characterizatio n of the channel with central and local decoding scheme, and limited ba c khaul capacity C is given by E b N 0 dec min = 1 λ o  E b N 0 d min  − 1 + (1 − λ o )  E b N 0 obl min ( r m )  − 1 S dec 0 =  E b N 0 dec min  − 2 λ o  S d 0  − 1  E b N 0 d min  − 2 + (1 − λ o )  S obl 0 ( r m )  − 1  E b N 0 obl min ( r m )  − 2 , (5-15) wher e the s uperscript ( · ) obl ( r m ) indicat es the low-SNR p arameters of t he oblivio us scheme (calculated usin g Proposition IV .5 with r m r eplacing C ), an d the not ation ( · ) d indicates the low-SNR parameters of the local d ecoding rate. Here, we are able to express the low-SNR parameters of the hybri d decoding scheme as functions of the low-SNR parameters of the respecti ve local decoding and the oblivious schemes. The latter can be further expressed in t erms o f the low-SNR parameters o f the no n- limited scheme according t o t he results o f Proposi tion IV .5. Exami ning (5-15) it is observed that for λ o = 0 (or C ≥ ˜ r m in (5-9)) the lo w-SNR parameters coin cide with those of the oblivious scheme (Proposition IV .5). Hence, all ocating reso urces to local decoding in the low- SNR regime is beneﬁcial when C is below a certain threshol d ˜ r m (see (5-10)). 20 4) High-SNR Characterizat ion: Similarly to the oblivious scheme, for any ﬁxed backhaul capacity C , the rate of th e partial local decoding scheme loses its mult iplexing g ain in the high-SNR regime. T his is easily concluded from the upper-bound (3-12). So we focus on the case i n which C i s allowed to increase wit h SNR and inter-cell interference is present (i.e. α > 0 ). From the time-sharing behavior of (5-5), and since i t is easily obs erved that local decoding does n ot att ain t he hi gh SNR parameters of the unrestri cted joi nt processing , it is concluded th at the high-SNR s olution t o the time sharing (5-5), is onl y obl ivious processing, which means r ∼ = r ∗ . Hence, i n this case no local decoding is performed and all resources are allocated to the oblivious scheme. Thus, in the high-SNR re gi me the two schemes are equiv alent and Cor . IV .6 holds for the th e partial local decoding scheme as well. It i s noted that when no inter-cell interference is p resent (i.e. α = 0 ), MCP i s not beneﬁcial and all resources can equiv alently be dev oted t o local decodi ng at the BSs. In order for the correspondi ng rate to maintain the high-SNR characterization of the original unli mited backhaul capacity setup, we should set C = ˆ R d (0) . Hence, C should scale as S d ∞ (log 2 P − L d ∞ ) (where S d ∞ , and L d ∞ are the high -SNR parameters of ˆ R d (0) ). A. Gaussi an Chan nels 1) The W yner Mod el: For the W yner model, local decoding can be performed i n three ways: decoding all three messages th at arriv e at the d estination; d ecoding only the strongest m essage, treating the rest as i nterference; and decoding onl y the si gnals from the adjacent cells. This giv es t he following rate for local decodi ng [8]: R d = max ( log 2  1 + (1 − β ) P 1 + ( β + 2 α 2 ) P  , min ( 1 2 log 2  1 + (1 − β )2 α 2 P 1 + β (1 + 2 α 2 ) P  , 1 3 log 2  1 + (1 + 2 α 2 )(1 − β ) P 1 + β (1 + 2 α 2 ) P  )) . ( 5-16) Substitutin g (5-16) i n (5-1) and (5-4) g iv es t he achie vable rate. T o consid er the low-SNR regime we apply the results o f Proposition V .2 whi ch approximate the low-S NR parameters of (5-5) for general rate expressions F ( r ) and ˆ R d (0) . N oting that th e low-SNR characterization of ˆ R d (0) is giv en by E b N 0 d min = log 2 ; S d 0 = 2 1 + 4 α 2 , ( 5-17) the low-SNR characterization of (5-5) is obtained by s ubstitut ing the low-SNR parameters of (4-25) and (5-17) in the general expressions (5-15), where r m = max { C , ˜ r m } and ˜ r m is the unique solut ion of 2 − ˜ r m (1 + ˜ r m log 2) = 2 α 2 1 + 2 α 2 . ( 5-18) Note t hat according to Proposition V .2 the t ime ratio dedicated to decoding at th e BSs in the l ow-S NR region is λ o = 1 − C r m . In additi on, examining (5-18) it is evident that ˜ r m is a 21 decreasing function of t he intra-cell int erference factor α . Therefore, it is concluded that in t he low-SNR region, decoding also at th e BSs i s b eneﬁcial if the backhaul capacity C i s b elow a certain threshold wh ich decreases with α . For example, when there is no inter- cell interference α = 0 then ˜ r m = ∞ and d ecoding only at the BSs i s optimal for any C . On the other hand, for α = 0 . 2 numerical calculation reveals th at ˜ r m ≈ 2 . 15 [bits]. Hence, i ncorporating decoding also at th e BSs is beneﬁcial when C . 2 . 1 5 [bit s]. 2) The S oft-Handoff Model: Simi larly to the local decoding scheme applied for t he W y ner model, here each user employs rate sp litting and divides its message into two parts: one that is decoded at t he RCP with p ower (1 − β ) P , and another t hat is decoded at the local cell-si te with power β P . As before there are two strategies for the cell-site to execute: to decode o nly its local user’ s message; or t o decode also the in terfering users’ messages emerging from the left neighborin g cell (see [8] Section III.D). Such approach allows decoding of messages with rate R sh d = max ( log 2  1 + (1 − β ) P 1 + ( β + α 2 ) P  , 1 2 log 2  1 + (1 − β )(1 + α 2 ) P 1 + β (1 + α 2 ) P  ) . ( 5-19) Repeating steps (5-1)-(5-3) with th e proper rate expressions for the SH model (expressions (5-19) and (4-26)) we get an achiev able rate similar to (5-4). As with the W yner model, by usin g time sharing between the e x treme ca ses β = 0 and β = 1 , we get an explicit achiev able rate expression R sh dec similar to (5-5) wit h t = min { C , R sh d (0) } . It is noted th at unli ke the W yner model, h ere r ∗ is explicitly giv en by r ∗ = C − R sh obl , where R sh obl is given by (4-26). T o consi der the low-SNR regime we apply the results of Proposition V .2. No ting that the low-SNR characterization of ˆ R sh d (0) is giv en by E b N 0 sh − d min = log 2 ; S sh − d 0 = 2 1 + 2 α 2 , (5-20) we obtain a similar result as with the W y ner model, with r m = max { C , ˜ r m } and w here ˜ r m is now the unique soluti on of 2 − ˜ r m (1 + ˜ r m log 2 ) = α 2 1 + α 2 . (5-21) Similar observations as those made for the non-fading W yner model are evident. B. F adi ng Chann els 1) The W yner Mod el: Introducin g fading, and adhering to the simp le scheme introduced in the previous section, decoding at an arbitrary BS (the cell i ndex is omitted) yi elds the fol lowing 22 rate R d − rf = max ( E log 2 1 + (1 − β ) 1 K | a | 2 P 1 +  β 1 K | a | 2 + α 2 ( 1 K | b | 2 + 1 K | c | 2 )  P !! , min ( 1 2 E log 2 1 + (1 − β ) α 2  1 K | b | 2 + 1 K | c | 2  P 1 + β  1 K | a | 2 + α 2 ( 1 K | b | 2 + 1 K | c | 2 )  P !! , 1 3 E log 2 1 + (1 − β )  1 K | a | 2 + α 2  1 K | b | 2 + 1 K | c | 2  P 1 + β  1 K | a | 2 + α 2 ( 1 K | b | 2 + 1 K | c | 2 )  P !! )) , ( 5-22) where the expectations are taken wi th respect to the fading coefﬁcient vectors a , b , and c . 3 Repeating steps (5-1)-(5-3) while s etting F ( r ) wit h (4-28) (or with (4-29) for a comp act yet subopti mal rate), we can get a si milar result as (5-4) for the fading channels wi th ﬁnite number of us ers per-cell K . Focusing on th e scenario o f a large number of users p er -cell K ≫ 1 with a ﬁxed total cell SNR P , it can be veriﬁed using t he st rong l aw of large nu mbers (SLLN) (see [8]) that (5-22) reduces to (5-16), and th at repeating steps (5-1)-(5-3) wh ile settin g F ( r ) = R rf − lk ( P (1 − 2 − r )) , we get the following achiev abl e rate: R rf − lk − ld = max β ( log 2 1 + (1 + 2 α 2 ) β P ( 1 − 2 − ( C − ˆ R d ( β )) ) 1 + (1 + 2 α 2 ) β P 2 − ( C − ˆ R d ( β )) ! + ˆ R d ( β ) ) . (5-23) Moreover , us ing time sharing between the two extreme β = 0 and β = 1 , we obtain R r f − l k − ld 2 = max r ≥ r ∗  t + ( C − t ) log 2 (1 + (1 + 2 α 2 ) P (1 − 2 − r )) − t log 2 (1 + (1 + 2 α 2 ) P (1 − 2 − r )) + r − t  , (5-24) where (via (4-31)) r ∗ = log 2 2 C + (1 + 2 α 2 ) P 1 + (1 + 2 α 2 ) P . (5-25) For the low-SNR regime, with a ﬁnite number of users per-cell ( K ﬁnite) we apply t he results of Propos ition V .2. The l ow-S NR characterization of ˆ R d − rf (0) is giv en by (see [1]) E b N 0 rf − d min = log 2 ; S rf − d 0 = 2 1 + 4 α 2 + 1 2 K , (5-26) while (4-32) is then also used in the general expressions (5-15), where r m = max { C, ˜ r m } is equal to t hat of (5-18). Similar conclusions as those of t he non-fading channels are evident. 3 It is noted that the expressions included in (5-22) can be rewritten as integrations over certain hypergeometric functions (see [8 ]). Si nce these integrals are numerically unstable, especially for large K they are omitted here. 23 C. The Soft-Han doff Model - F ading Channels Introducing fading, and adhering to th e simple scheme introdu ced in the previous section, decoding at an arbitrary BS yi elds the following rate R sh d − rf = max ( E log 2 1 + (1 − β ) 1 K | a | 2 P 1 +  β 1 K | a | 2 + α 2 1 K | b | 2 )  P !! , 1 2 E log 2 1 + (1 − β ) α 2 1 K | b | 2 P 1 + β  1 K | a | 2 + α 2 1 K | b | 2  P !! ) , ( 5-27) where the expectations are taken wi th respect to the fading coefﬁcient vectors a and b . As wi th the W yner mod el, we repeat steps (5-1)-(5-3) setti ng F ( r ) with speciﬁc expressions providing achiev able rates for severa l cases of interest: (a) ﬁnite num ber of users K - usin g (4-28) (or wit h (4-29) for a compact yet s uboptimal rate) while repl acing H N with the SH channel transfer m atrix H sh N ; (b) upper bounds for ﬁnite num ber of u sers K - us ing (4-34); and (c) TDMA with Rayleigh fading channels and α = 1 - using the exact rate expression (4-33). Focusing on the s cenario of a large num ber of users per-cell K ≫ 1 w ith ﬁxed to tal cell SNR P , it is can be veriﬁe d using t he SLLN (see [8]) that (5-27) reduces to (5-19), and that b y repeating steps (5-1)-(5-3) while setting F ( r ) = R sh rf − lk ( P (1 − 2 − r )) , we g et si milar expressions as t hose derive d for the W y ner mod el while replacing the W yner array power gain (1 + 2 α 2 ) with th e power gain of the SH array (1 + α 2 ) in expressions (5-23), (5-24), and (5-25), respectively . Finally , to consider the low-S NR regime we app ly the results o f Propos ition V .2. Noticing that the l ow-S NR characterization o f ˆ R sh d − rf (0) is giv en by E b N 0 sh − rf − d min = log 2 ; S sh − rf − d 0 = 2 1 + 2 α 2 + 1 K , (5-28) we hav e that th e low-SNR characterization i s obtained by using the low-SNR characterization of the l ocal decoding (5-28) and of the o blivious processi ng (4-36) in the general expression (5-15), where r m = max { C , ˜ r m } is equal to that o f (5-21). V I . N U M E R I C A L R E S U L T S In this section we demo nstrate the ef fects of limi ted-capacity backhaul links by sever al numerical examples. For the sake of conciseness , on ly the W yner mod el is considered since similar conclusi ons apply for the SH model. Achiev able rates of the (a) o blivious scheme, (b) local-decoding s cheme, and (c) unlim ited setup, are plotted as functio ns of the inter-cell interference factor α for t otal cell power P = 1 0 [dB] and Gaussian (non -fading) channels, in Figures 2 and 3 for backhaul l ink capacity of C = 3 [bits/channel use] and C = 6 , respectiv ely . Upon examining the ﬁgures the deleterious ef fects of limited backhaul are re vealed. In add ition, the beneﬁts of l ocal decoding are evident for int erference levels below a certain th reshold which decreases wi th increasing va lues of C . 24 In particul ar , in Figure 2 wh ere C = 3 , the local decoding rate achiev es th e upper bound of the limited backhaul rate for interference leve ls below a certain threshold. This range reduces to a sin gle point α = 0 for large va lues of C (e.g. Figure 3 where C = 6 ), where local decoding at th e BSs alone is optimal sin ce no inter-ce ll interference is present. On the other hand the oblivious approach cannot achiev e the upper bou nd for ﬁnite values of C . Introducing Rayleigh fading channels, the s ame rates are p lotted for seve ral values of the number of users per-cell K in Figures 4 and 5 for backhaul link capacity of C = 3 [bits/channel use] and C = 6 respectively . Exami ning the ﬁgures, si milar observ ations as those for the Gaussian channels are e vident. As with the unl imited setup stu died in [9], th e rates i n general increase with th e number of users per-cell K , in the presence of fading. In Figures 6 and 7, the upper-bound (which is the minimum between the unrestricted backhaul rate and the backhaul capacity) and the achie vable rates of the oblivious and local- decoding schemes are plott ed as functions of the total-cell power P wi th C = 6 and α = 0 . 15 , for Gaussian and ﬂat Rayleigh fading channels respectiv ely . For Gauss ian channels, the curves of the oblivious and the l ocal decoding schemes are very close sin ce the interference l e vel is small, so decoding at the BSs brings marginal im provement. It i s also obs erved that the rate of the oblivious scheme appro aches the upp er bound for low SNR values wh ere C is much l ar ger than t he unli mited rate, and for large SNR values where the u nlimited rate is much larger than C . Similar behavior is observed for fading channels. In Figures 8 and 9, t he upper bound and the achiev able rates are plott ed as functions of the backhaul li nk capacity with P = 10 and α = 0 . 4 , for Gaussi an and ﬂat Rayleigh fading channels respectiv ely . For both Gaussian and fading channels, it is observed that the rates of both schemes approach the upper bo und for low values of C , where the u nlimited rates are much high er th an C , or for l ar ge values of C , where C i s mu ch higher th an the unl imited rates. Moreover , the beneﬁt of the local decoding scheme is evident where in fact its rate achieves the upper boun d C , below a certain threshold. As before, sim ilar observations are m ade for the fading channels. T o conclude t his section we verify t he low-SNR regime analyti cal results derived for the oblivious and local decoding schemes wit h some num erical results derived for the W yner setup with Gauss ian channels. In Figures 10-12 the spectral efﬁ ciencies of the per-cell s um- rates of the (a) unlim ited setup wi th opti mal MCP , (b) the local decoding scheme, and (c) the oblivious scheme, are plott ed for C = 2 , 4 and 6 [bits] respectively . For t he two schemes, both the exact rates and low-SNR approximatio ns (based on Propos itions IV .5 and V .2) are plotted. Examining the curves it is observed that whil e t he approximated m inimum E b N 0 fairly matches the n umerical results, the approxim ated low-S NR slope i s somewhat optimisti c when compared to the exact curves. Moreover , the beneﬁt of the local decodi ng scheme is evident when the backhaul capacity C is sm all. V I I . C O N C L U D I N G R E M A R K S In thi s paper we hav e consid ered symm etric cellul ar model s with limited backhaul capacity . Simple and t ractable achiev able rates hav e been derived for the case of cell-sites that use sign al processing alone, and also when comb ined with local decoding. Both schemes cons idered no 25 network planning, so in ter- cell interference dominates. Clos ed form e xpressions ha ve been dev elo ped for both the classical W y ner mod el and t he SH model. Addit ional explicit l ow-S NR approximations for the achiev able rates have als o been d eriv ed. The rate at which the backhaul capacity should scale with SNR, in order for the v arious schemes to maintain their original hig h- SNR characterization, have also been derived. All results in this paper have i ncluded analysis for both Gaussi an and Rayleigh fading channels. Nu merical calculati ons revea l that un limited optimal joint processi ng performance can be clos ely app roached with a rather limi ted backhaul capacity , and that addition al lo cal decoding is beneﬁcial o nly for low inter-cell i nterference lev el s. A P P E N D I X I P R O O F O U T L I N E O F P R O P O S I T I O N I V . 1 The proof of Propositi on IV .1 is based on the proof of Theorem 3 from [13]. W e need t he following lemm a. Lemma I.1 Generalized Markov Lemma Let P A S ,Y S | H ( a S , y S | h ) = P Y S | H ( y S | h ) Y j ∈S P A j | Y j ,H ( a j | y j , h ) . (I.1) Given r andomly g enerated y S accor ding to P Y S | H , for every j ∈ S , randomly an d independently generate N j ≥ 2 nI ( A j ; Y j | H ) vectors ˜ a j accor ding to Q n t =1 P A j | H (˜ a j ( t ) | h ( t )) , and index them by ˜ a ( v ) j ( 1 ≤ v ≤ N j ). Then t her e e xist |S | functions v ∗ j = φ j ( y j , ˜ a (1) j , . . . , ˜ a ( N j ) j ) taking values in [1 . . . N j ] , such that fo r any ǫ > 0 and sufﬁciently lar ge n , Pr(( { a ( v ∗ j ) j } j ∈S , y S ) ∈ T ǫ ( h )) ≥ 1 − ǫ. (I.2) Pr oof: Lemma I.1 is Lemm a 3.4 (Generalized M arkov Lemma) in [24]. A. Code cons truction: For ever y channel realizatio n h , d etermine the m aximizing π . Fix δ > 0 and then I) For every user k , within th e j th cell, • Random ly choos e 2 nR j,k vectors x j,k , with probability P X j,k ( x j,k ) = Q t P X j,k ( x j,k ( t )) . • Ind ex these vectors by M j,k where M j,k ∈ [1 , 2 nR j,k ] . II) For the compressor at the cell-sites For e very cell-si te j and every channel realizatio ns matrix H • Random ly generate 2 n [ ˆ R j − C j ] vectors u j of length n according to Q t P U j | H ( u j ( t ) | h ( t )) . • Repeat the last step for s j = 1 , . . . , 2 nC j , deﬁne the resulti ng set of u j of each repetition by S s j . • Ind ex all the generated u j with z j ∈ [1 , 2 n ˆ R j ] . W e will interchangeably use the notation S s j for the set of vectors u j as well as for the s et of t he correspondi ng z j . • No tice t hat t he m apping between the indices z j and th e vectors u j depends on h . So we will write u j ( z j , h ) to denote u j which is indexed by z j for some speciﬁc h . 26 B. Encoding: Let M = ( M j,k ) N ,K j =1 ,k =1 be the joint m essages to b e sent. The transm itters then s end the corresponding ( x j,k ) N ,K j =1 ,k =1 to the channel. C. Pr ocessin g at the cell-sit es: The j th cell-site chooses any of th e z j such that  u j ( z j , h ) , y j  ∈ T j ǫ ( h ) , (I.3) where T j ǫ ( h ) ,      u j , y j : ∀ u ∈ U j , h ∈ H : 1 n   N ( u, h | u j , h ) − P U j | H ( u | h ) N ( h | h )   < ǫ |U j | ∀ y ∈ Y j , h ∈ H : 1 n   N ( y , h | y j , h ) − P Y j | H ( y | h ) N ( h | h )   < ǫ |Y j | u ∈ U j , y ∈ Y j , h ∈ H : 1 n   N ( u, y , h | u j , y j , h ) − P U j ,Y j | H ( u, y | h ) N ( h | h )   < ǫ |Y j ||U j |      . (I.4) The e vent where no such z j is found is deﬁned as the error e vent E 1 . After decidi ng o n z j the cell-site transmit s s j , which ful ﬁlls z j ∈ S s j , to th e RCP th rough the lossl ess link. D. Decoding (at t he RCP): The destination retrieves s N , ( s 0 , . . . , s N − 1 ) from the lossless link s. It t hen ﬁnds the set of ind ices ˆ z N , { ˆ z 1 , . . . , ˆ z N } of the comp ressed vectors ˆ u N and the messages M which satisfy (  x N , K ( M N , K ) , ˆ u N ( ˆ z N , h )  ∈ T 3 ǫ ( h ) ˆ z N ∈ S s 0 × · · · × S s N − 1 (I.5) where T 3 ǫ is deﬁned in t he st andard way , as (I.4). If t here is no s uch ˆ z N , ˆ M N , K , or if there is more than one, the destinati on choos es one arbit rarily . Deﬁne error E 2 as the event where ˆ M N , K 6 = M N , K . Correct decoding means that the destin ation decides ˆ M N , K = M N , K . An achiev abl e rate region R N , K was deﬁned as when the RCP receives the t ransmitted messages wit h an error probabilit y which is made arbitrarily sm all for sufﬁciently l ar ge blo ck length n . E. Err or analysis The error probabi lity is upper bounded by Pr { error } = Pr ( E 1 ∪ E 2 ) ≤ Pr( E 1 ) + Pr( E 2 ) , (I.6) where I) E 1 is the event t hat no u j ( z j , h ) is jointly typi cal with y j , and II) E 2 is the event t hat there is a decoding error ˆ M N , K 6 = M N , K . Next, we will upper bound the probabi lities o f the individual error ev ent s b y arbitrarily small ǫ . 27 1) E 1 : According to Lem ma I.1, the probabili ty Pr { E 1 } can be made as sm all as desired, for n su fﬁ ciently large, as long as ˆ R j > I ( U j ; Y j | H ) . (I.7) 2) E 2 : Cons ider the case where ˆ M L , Z 6 = M L , Z and ˆ z S 6 = z S , where S , L ⊆ N and Z ⊆ K . There are 2 n [ P j ∈L ,k ∈Z R j,k + P i ∈S [ ˆ R i − C i ]] such vectors, and the probabili ty of ( x N , K ( ˆ M N , K ) , u N ( ˆ z N )) to be jointly typical is up per bounded by [13] 2 n [ h ( X N , K ,U N | H ) − h ( U S C ,X {N , K} C | H ) − P i ∈S h ( U i | H ) − P j ∈L ,k ∈Z h ( X j,k )+ ǫ ] , where h functions here also as the diff erential ent ropy . Thus Pr { E 2 } can be m ade arbit rarily small as long as the rate region R N , K for all L , S ⊆ N and Z ⊆ K sati sﬁes X j ∈L ,k ∈Z R j,k < X i ∈S [ C i − ˆ R i + h ( U i | H ) − h ( U i | H , X N , K )] − h ( X L , Z | X {L , Z } C , U S C , H ) + X j ∈L ,k ∈Z h ( X j,k | X {L , Z } C , H ) = X i ∈S [ C i − I ( Y i ; U i | X N , K , H )] + I ( U S C ; X L , Z | X {L , Z } C , H ) , (I.8) where (I.8) is due to of the foll owing equalities: h ( U i | Y i ) = h ( U i | Y i , X N , K ) , h ( X L , Z ) = h ( X L , Z | X {L , Z } C ) . h ( X L , Z , U S | X {L , Z } C , U S C ) = h ( X L , Z | X {L , Z } C , U S C ) + X i ∈S h ( U i | X N , K ) . Equation (I.8) completes the proof. A P P E N D I X I I P R O O F O F L E M M A I V . 4 W e prove that at least one S which m inimizes lim N →∞ 1 N I ( X N , K ; U S ) = lim N →∞ 1 N log 2 det( I + P ′ H S H ∗ S ) , when |S | = f ( N ) , ( f : R + 7→ R + , lim N →∞ f ( N ) N = λ , 0 ≤ λ ≤ 1 ), is compo sed of onl y consecutiv e indices. Following the m ethod us ed i n [25] to derive a lower bound on t he capacity of the Gaussian erasure channel, the proof here u ses an analogy between th e mult i-cell setup and an i nter-symbol interference (ISI) channel, combined with a recently reported relationship between the MM SE and the mutual inform ation [26]. 28 Pr oof: Denote by E i , the MM SE i ncurred when estim ating H i X N , K from U N , h . Further denote by E i ( S ) , the MMSE incurred when esti mating H i X N , K from U S , h . Naturally ∀ h, S ⊆ N , i ∈ S : E i ( S ) ≥ E i , (II.1) and also i / ∈ S : E i ( S ) = 0 . (II.2) Next, we use th e following relationship between the MMSE and the m utual informatio n [26], to writ e d dP I ( X N ; U S | H ) = N − 1 X i =0 E i ( S ) . (II.3) From (II.1) and the ergodicity of the channel, we can write N − 1 X i =0 E i ( S ) ≥ f ( N ) N N − 1 X i =0 E i . (II.4) Combining (II.3) and (II.4) yi elds I ( X N ; U S | H ) ≥ f ( N ) N Z P ′ 0 N − 1 X i =0 E i dP = f ( N ) N I ( X N ; U N | H ) . (II.5) On the ot her hand, in the asympto tic regime, for consecutive indices set S ( c ) , where lim N →∞ |S ( c ) | N = λ , we hav e lim N →∞ 1 N X i ∈S ( c ) E i ( S ( c ) ) = λ lim N →∞ 1 N N − 1 X i =0 E i . (II.6) This is because the equiv alent ISI channel is stationary , and since the right hand side of (II.6) exists. By integrating b oth sides of equation (II.6) we get th at lim N →∞ 1 N I ( X N ; U S ( c ) | H ) = λ lim N →∞ 1 N I ( X N ; U N | H ) . (II.7) Equation (II.7) t ogether with (II.5) proves the l emma. A P P E N D I X I I I P R O O F O F T H E AC H I E V A B L E R A T E O F T H E S H M O D E L W I T H L I M I T E D BA C K H AU L A N D G A U S S I A N C H A N N E L S , E Q U A T I O N ( 4 - 2 6 ) Using arguments similar to these used for the W yner setup, the per -cell sum-rate of the limited soft-handoff setup is given by Propos ition IV .3, with F ( r ∗ ) = R sh nf ( P (1 − 2 − r ∗ ) where R sh nf is the rate o f the unli mited setup given i n (3-4). Due to the explicit simple form of F ( r ∗ ) , the ﬁxed point equatio n (4-7) reduces t o the fol lowing quadratic equation: (1 + P 2 − C )(1 + α 2 P 2 − C ) x 2 −  1 + (1 + α 2 ) P + 2 α 2 P 2 2 − C  x + α 2 P 2 = 0 (III.1) 29 where x = 2 C − r ∗ , and its roots are given by x 1 , 2 = 1 + (1 + α 2 ) P + 2 α 2 2 − C P 2 ± p 1 + 2(1 + α 2 ) P + ((1 − α 2 ) 2 + 4 α 2 2 − C ) P 2 2(1 + 2 − C P )(1 + α 2 2 − C P ) . (III.2) For x 1 we h a ve t he following set of inequali ties x 1 ≥ 1 + (1 + α 2 )2 − C P + 2 α 2 2 − 2 C P 2 + p 1 + 2(1 + α 2 )2 − C P + ((1 − α 2 ) 2 + 4 α 2 ) 2 − 2 C P 2 2(1 + 2 − C P )(1 + α 2 2 − C P ) = 1 + (1 + α 2 )2 − C P + 2 α 2 2 − 2 C P 2 + p 1 + 2(1 + α 2 )2 − C P + (1 + α 2 ) 2 2 − 2 C P 2 2(1 + 2 − C P )(1 + α 2 2 − C P ) = 1 + (1 + α 2 )2 − C P + 2 α 2 2 − 2 C P 2 + q (1 + (1 + α 2 )2 − C P ) 2 2(1 + 2 − C P )(1 + α 2 2 − C P ) = 1 + (1 + α 2 )2 − C P + α 2 2 − 2 C P 2 (1 + 2 − C P )(1 + α 2 2 − C P ) = 1 . (III.3) Hence, choosing + in (III.2) yi elds a valid solut ion with r ∗ which i s also smal ler than t he backhaul capacity C for all values o f P , C and α (i.e. x 1 ≥ 1 , ∀ P , C , α ). Recalling that the rate equals C − r ∗ (see (4-7)) completes the deriv ation. A P P E N D I X I V P R O O F O F T H E AC H I E V A B L E R A T E O F T H E S H M O D E L W I T H L I M I T E D BA C K H AU L A N D F A D I N G C H A N N E L S , E Q UA T I O N ( 4 - 3 4 ) W e start b y ob serving that replacing F ( r ∗ ) in (4-6) with an upper bo und F ub ( r ) ≥ F ( r ∗ ) where F ub (0) = 0 , provides a valid solution to (4-7), which is als o an upper bound on F ( r ∗ ) . Setting F ( r ∗ ) = R sh ub − rf ( P (1 − 2 − r ∗ )) where R sh rf − ub is given in (3-9) and so lving the ﬁxed point equation (4-7), we get t he following quadratic equation  1 + P (1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K  x 2 −  1 + P (1 + α 2 ) + 2 α 2 P 2 2 − C /K  x + P 2 α 2 /K = 0 , (IV .1) where x = 2 C − r ∗ , and its roots are given by x 1 , 2 = 1 + P (1 + α 2 ) + 2 P 2 α 2 2 − C /K ± q (1 + P (1 + α 2 )) 2 − 4 P 2 α 2 (1 − 2 − C ) /K 2 (1 + P ( 1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K ) . (IV .2) 30 For x 1 we h a ve t he following set of inequali ties x 1 ≥ 1 + P (1 + α 2 ) + 2 P 2 α 2 2 − C /K ± q (1 + P (1 + α 2 )) 2 − 4 P 2 α 2 (1 − 2 − C ) 2 (1 + P ( 1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K ) = 1 + P (1 + α 2 ) + 2 P 2 α 2 2 − C /K + p 1 + 2 P (1 + α 2 ) + P 2 ((1 − α 2 ) 2 + 4 α 2 2 − C ) 2 (1 + P (1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K ) ≥ 1 + P (1 + α 2 )2 − C + 2 P 2 α 2 2 − 2 C /K + p 1 + 2 P (1 + α 2 ) + P 2 ((1 − α 2 ) 2 + 4 α 2 ) 2 − C 2 (1 + P (1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K ) ≥ 1 + P (1 + α 2 )2 − C + 2 P 2 α 2 2 − 2 C /K + p 1 + 2 P (1 + α 2 )2 − C + P 2 (1 + α 2 ) 2 2 − 2 C 2 (1 + P (1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K ) = 1 + P (1 + α 2 )2 − C + 2 P 2 α 2 2 − 2 C /K + q (1 + P (1 + α 2 )2 − C ) 2 2 (1 + P (1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K ) = 2  1 + P (1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K  2 (1 + P (1 + α 2 )2 − C + P 2 α 2 2 − 2 C /K ) = 1 . (IV .3) Hence, cho osing + in (IV .2) yields a valid solut ion with r ∗ which i s also small er t han th e backhaul capacity C for all values o f P , C and α (i.e. x 1 ≥ 1 , ∀ P , C , α ). Recalling that the rate upper b ound equals C − r ∗ (see (4-7)) compl etes the deriva tion. A P P E N D I X V D E R I V A T I O N O F E X PR E S S I O N (5-5) The deriv ation is based on time-sharing between the point ( t, t ) and the con ca ve curve ( F ( r ) + r, F ( r )) (usin g power P in both techniques). Th e ﬁrst poi nt is achieve d by us ing local decoding at the cell-site, while the second is achieved by using oblivious processing at t he cell-sites and RCP decoding. Using only local decoding is opt imal when R d (0) = C . When R d (0) < C , it i s worthwhile to us e tim e sharing with s ome point ( F ( r ′ ) + r ′ , F ( r ′ )) . Thi s means ( λt + ( 1 − λ )( F ( r ′ ) + r ′ ) = C λt + ( 1 − λ ) F ( r ′ ) = R dec . (V .1) From ﬁrst equation of (V .1), we have λ = C − ( F ( r ′ ) + r ′ ) t − ( F ( r ′ ) + r ′ ) , (V .2) and assign ing this value back to the second equation of (V .1) we obtain R = C − ( F ( r ′ ) + r ′ ) t − ( F ( r ′ ) + r ′ ) t + t − C t − ( F ( r ′ ) + r ′ ) F ( r ′ ) = t − C − t t − ( F ( r ′ ) + r ′ ) t + t − C t − ( F ( r ′ ) + r ′ ) F ( r ′ ) = t + ( C − t ) F ( r ′ ) − t F ( r ′ ) + r ′ − t . (V .3) 31 Next, we would like to opti mize over r ′ , such that we get the maximal rate. Considering th at 0 ≤ λ ≤ 1 , r ′ must be larger than r ∗ , thus limiting the opt imization range. R E F E R E N C E S [1] O. Somekh, O. Simeone, Y . B ar-Ness, A. M. Haimovich, U. Spagnolini, and S. Shamai (Shitz), Distributed Antenna Systems: Open Arc hitecture f or Futur e W ir eless Communications . Auerbach P ublications, CRC Press, May 2007, ch. An Information Theoretic V iew of Distri buted Antenna Processing in Cell ular S ystems. [2] S. S hamai, O. Somekh, and B. M. Z aidel, “Multi-cell communications: An information theoretic perspectiv e, ” in Pr oceedings of the Joint W orkshop on Communications and Coding (JWCC’04) , Donnini, F lorence, Italy , Oct. 2004. [3] O. Somekh, B. M. Z aidel, and S. Shamai, “Spectral ef ﬁ ciency of joint multiple cell-site processo rs for randomly spread DS-CDMA systems, ” IEEE T rans. Inform. Theory , vol. 52, no. 7, pp. 2625–2637, Jul. 2007, to appear . [4] E. Aktas, J. Ev ans, and S. Hanly , “Di stributed decoding in a cellular multiple a ccess channel, ” in Pr oc. IEEE International Symposium on Inform. T heory (ISIT ’04) , Chicago, Illinois, Jun. 27-Jul. 2 2004, p. 484. [5] O. S hental, A. J. W eiss, N. Shental, and Y . W eiss, “Generalized belief propagation receiv er for near optimal detection of two-dimension al channels with memory , ” in Pr oc. Inform. Theory W orkshop (ITW’04) , S an Antonio, T exas, Oct. 24-29 2004. [6] P . Marsch and G. Fett weis, “ A f rame work for optimizing the uplink performance of distributed antenna systems under a constrained backhaul, ” in Proc. of the IEEE International Confer ence on Communications (ICC’07) , Glasgo w , Scotland, Jun. 24-28 2007. [7] A. W yner , “Shannon theoretic approach to a Gaussian cellular multiple access channel, ” IE EE T rans. Inform. Theory , vol. 40, no. 6, pp. 1713–1727, Nov 1994. [8] S. Shamai and A. W yner , “Information-theoretic considerations for symmetric cellular, multiple-access fading channels - part I, ” IEEE T rans. Inform. Theory , vol. 43, no. 6, pp. 1877–1894, Nov 1997. [9] O. Somekh and S. Shamai, “Shanno n-theoretic approach to a Gaussian cellular multi-access channel with fading, ” IEE E T rans. Inform. T heory , vol. 46, no. 4, pp. 1401–1425 , July 2000. [10] A. Sanderovich, S . Shamai, Y . Steinberg, and G. Kramer, “Communication via decentralized processing, ” IEEE T rans. Inform. Theory , vol. 54, no. 7, July 2008. [11] ——, “Communication via decentralized processing , ” in P r oc. of IEE E I nt. Symp. Info. Theory (ISIT2005) , Adelaide, Australia, Sep. 2005, pp. 1201–1205. [12] A. Sanderovich, S . Shamai, Y . S teinberg, and M. P eleg, “Decentralized receive r in a MIMO system, ” in Proc . of IEEE Int. Symp. Info. T heory (ISIT’ 06) , Seatt le, W A, July 2006, pp. 6–10. [13] A. Sanderovich, S. Shamai, and Y . Steinberg, “Decentralized receiv er in a MIMO system, ” Submitted to IEEE T rans. Inform. Theory . [O nline]. A vailable: http://arxiv .org/abs/0710.011 6v1 [14] S. V erd ´ u, “Spectral efﬁciency in the wideband regime, ” vol. 48, no. 6, pp. 1329–1343, jun 2002. [15] S. V . Hanly and P . A. W hiting, “Information-theoretic capacity of multi-receiv er networks, ” T elecommun. Syst. , vol. 1, pp. 1–42, 1993. [16] N. Le vy , O. Somekh, S. S hamai, and O. Zeitouni, “On certain large random hermitian j acobi matrices with applications to wireless communications, ” Submitted to the IEEE T rans. Inform. T heory , 2007. [17] A. Lozano, A. Tu lino, and S. V erd ´ u, “High-SNR power offset in multi-antenna communications, ” vol. 51, no. 12, pp. 4134–4 151, Dec. 2005. [18] O. Somekh, B. M. Zaidel, and S. Shamai (Shitz), “Sum-rate characterization of multi-cell processing, ” in Pro ceedings of the Canadian workshop on i nformation theory (CWIT’ 05) , McGill U ni versity , Montreal, Quibec, Canada, Jun. 5–8, 2005. [19] ——, “Sum rate characterization of joint multiple cell-site processing, ” IEEE Tr ansactions on Information T heory , Dec. 2007. [20] S. Ji ng, D. N. C. Tse, J. Hou, J. Soriaga, J. E. S mee, and R. Padov ani, “Do wnlink macro-div ersity in cellular netwo rks, ” in the IE EE Intl. Symp. on Information Theory (ISIT’07) , Nice, France, Jun. 2007, pp. 1–5. [21] A. Narula, “Information theoretic analysis of multiple-antenna transmission div ersity , ” PhD Thesis, Massachusetts Institute of T echnology (MIT), Boston, MA, June 1997. [22] Y . Lifang and A. Goldsmith, “S ymmetric rate capacity of cellular systems with cooperati ve base stations, ” in Pr oceedings of Globecom 2006 , 2006. [23] T . M. Cov er and J. A. Thomas, Elements of Information theory . John Wile y & Sons, Inc., 1991. [24] T . S. Han and K . Kob ayashi, “ A uniﬁed achiev able rate region for a general class of multiterminal source coding systems, ” IEEE T rans. Inform. Theory , vol. IT -26, no. 3, pp. 277–288, May 1980. 32 [25] A. M. T ulino, S. V erd ´ u, G. Caire, and S. Shamai, “Capacity of the Gaussian erasure chann el, ” Submitted to IEEE T rans. Inform. Theory , t o be presented in part at ISIT 2007. [26] D. Guo, S. Shamai, and S. V erd ´ u, “Mutual information and minimum mean-square error in Gaussian channels, ” IEE E T rans. Inform. T heory , vol. 51, no. 4, pp. 1261–1282 , April 2005. 33 User 1 Cell 1 Cell-site 1 Cell-site 2 Cell-site 0 Cell-site N RCP C C C C C C Joint Processing a c a ab Fig. 1. Simple ﬁnite circular symmetric cellular model, with limited ﬁnite capacity to a central processing unit. One user is also drawn to demonstrate the model. 0 0.2 0.4 0.6 0.8 1 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4 α R [bits/channel use] Unrestricted Oblivious, C=3 C=3, w dec Fig. 2. The achie vab le rate for limited capacity C = 3 bits per channel use and oblivio us processing (solid line) or combined wi th local decoding (circle marker), as compared with t he unrestricted backhaul rate (dashed li ne), as a f unction of the interference level α . The signal to noise ratio is P = 10 dB, and the channel is additive white Gaussian. 34 0 0.2 0.4 0.6 0.8 1 2.8 3 3.2 3.4 3.6 3.8 4 α R [bits/channel use] Unrestricted Oblivious, C=6 C=6, w dec Fig. 3. The achie vab le rate for limited capacity C = 6 bits per channel use and oblivio us processing (solid line) or combined wi th local decoding (circle marker), as compared with t he unrestricted backhaul rate (dashed li ne), as a f unction of the interference level α . The signal to noise ratio is P = 10 dB, and the channel is additive white Gaussian. 35 0 0.2 0.4 0.6 0.8 1 2 2.5 3 3.5 4 4.5 5 5.5 α R [bits/channel use] Unrestricted fading K=1 Unrestricted fading K=5 Unrestricted fading K= ∞ Oblivious, C=3 fading K=1 Oblivious, C=3 fading K=5 Oblivious, C=3, fading K= ∞ fading, C=3, K= ∞ , w dec Fig. 4. The achiev able rate for limited capacity C = 3 bits per channel use and oblivious processing (solid lines) or combined with local decoding (circle marker), as compared with the unrestricted backhaul rate (dashed lines), as a function of the interference l ev el α . The ave rage per-cell sum signal to noise ratios is P = 10 dB, and the channel is Rayleigh ﬂat fading. T hree access protocols are plotted, with TDMA (triangular marker), with WB f or ﬁv e users (square) and with W B for inﬁnitel y many users (pluses). 36 0 0.2 0.4 0.6 0.8 1 2.5 3 3.5 4 4.5 5 α R [bits/channel use] Unrestricted fading K=1 Unrestricted fading K=5 Unrestricted fading K= ∞ Oblivious, C=6 fading K=1 Oblivious, C=6 fading K=5 Oblivious, C=6, fading K= ∞ fading, C=6, K= ∞ , w dec Fig. 5. The achiev able rate for limited capacity C = 6 bits per channel use and oblivious processing (solid lines) or combined with local decoding (circle marker), as compared with the unrestricted backhaul rate (dashed lines), as a function of the interference l ev el α . The ave rage per-cell sum signal to noise ratios is P = 10 dB, and the channel is Rayleigh ﬂat fading. T hree access protocols are plotted, with TDMA (triangular marker), with WB f or ﬁv e users (square) and with W B for inﬁnitel y many users (pluses). 37 0 5 10 15 20 25 30 1 2 3 4 5 6 7 SNR [dB] R [bits/channel use] Cut−set Oblivious, C=6 C=6, w dec Fig. 6. The achie vab le rate for limited capacity C = 6 bits per channel use and oblivio us processing (solid line) or combined with local decoding (circle marker), as compared wi th the upper bound (minimum of unrestricted backhaul rate and the backha ul rate - dashed line), as a function of t he signal to noise ratio P [dB], and the channel is additi ve white Gaussian. The interference level is α = 0 . 15 . 38 10 15 20 25 30 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 SNR [dB] R [bits/channel use] Upper bound, fading K=1 Upper bound, fading K=5 Upper bound, fading K= ∞ Oblivious, C=6 fading K=1 Oblivious, C=6 fading K=5 Oblivious, C=6, fading K= ∞ fading, C=6, K= ∞ , w dec Fig. 7. The achiev able rate for limited capacity C = 6 bits per channel use and oblivious processing (solid lines) or combined wit h local decodin g (circle marke r), as compared with the upper bound (minimum of unrestricted backhaul rate and the backh aul rate - dashed lines), as a function of the averag e signal to noise ratio P [ dB], when the channel i s R ayleigh fading. The interference lev el is α = 0 . 15 . Three access protocols are plotted, with TDMA (triangular marker), with WB for ﬁv e users (square) and with WB f or inﬁnitely many users (pluses). 39 0 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 C [bits per channel use] R [bits/channel use] Upper bound Oblivious, C=6 C=6, w dec Fig. 8. T he achiev able rate f or l imited capacity C bits per channel use and oblivious processing (solid line) or combined with local decoding (circle marker), as compared with the upper bound (minimum of unrestricted backhaul rate and the backhaul rate - dashed line), as a function of the backhaul capacity [bits per channel use]. The chann el is additiv e white Gaussian with SNR P = 10 dB, and the interference level is α = 0 . 4 . 40 0 1 2 3 4 5 6 7 8 0 0.5 1 1.5 2 2.5 3 3.5 4 C [bits per channel use] R [bits/channel use] Upper bound, fading K=1 Upper bound, fading K=5 Upper bound, fading K= ∞ Oblivious, fading K=1 Oblivious, fading K=5 Oblivious, fading K= ∞ fading, K= ∞ , w dec Fig. 9. The achiev able rate for limited capacity C bits per channel use and obliviou s processing (solid lines) or combined with local decoding (circle marker), as compared with the upper bound (minimum of unrestricted backhaul rate and the backhaul rate - dashed lines), as a function of the backhaul capacity [bits per channel use]. T he channel is Rayleigh fading with average S NR P = 10 dB, and the interference lev el α = 0 . 4 . Three access protoco ls are plotted, wit h TDMA (triangular marker), wit h WB for ﬁv e users (square) and with WB for inﬁnitely many users (pluses). 41 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 EbN0 [dB] Per−Cell Sum−Rate [bits/sec/Hz] Unlimited Capacity Local Decoding (Approx) Local Decoding Oblivious (Approx) Oblivious Fig. 10. Low-SNR region of t he per-cell sum-rate, as a function of E b N 0 for oblivious processing (solid line) and local decoding (plus marker) along wit h the corresponding approximation (dashed) and the upper bound (upper dashed l ine). T he channel is additi ve white Gaussian, with interference l ev el α = 0 . 2 , and backhaul capacity of 2 [bits/channel]. 42 −2 −1.5 −1 −0.5 0 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 EbN0 [dB] Per−Cell Sum−Rate [bits/sec/Hz] Unlimited Capacity Local Decoding (Approx) Local Decoding Oblivious (Approx) Oblivious Fig. 11. Low-SNR region of t he per-cell sum-rate, as a function of E b N 0 for oblivious processing (solid line) and local decoding (plus marker) along wit h the corresponding approximation (dashed) and the upper bound (upper dashed l ine). T he channel is additi ve white Gaussian, with interference l ev el α = 0 . 2 , and backhaul capacity of 4 [bits/channel]. −2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 EbN0 [dB] Per−Cell Sum−Rate [bits/sec/Hz] Unlimited Capacity Local Decoding (Approx) Local Decoding Oblivious (Approx) Oblivious Fig. 12. Low-SNR region of t he per-cell sum-rate, as a function of E b N 0 for oblivious processing (solid line) and local decoding (plus marker) along wit h the corresponding approximation (dashed) and the upper bound (upper dashed l ine). T he channel is additi ve white Gaussian, with interference l ev el α = 0 . 2 , and backhaul capacity of 6 [bits/channel].

Uplink Macro Diversity of Limited Backhaul Cellular Network

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment