Cellular Systems with Full-Duplex Compress-and-Forward Relaying and Cooperative Base Stations

Cellular Systems with Full-Duple x Compress-and-F orwa rd Relaying and Coo perati v e Base Stations Oren Somekh ∗ , Osv aldo Simeone † , H. V incent Poor ∗ , and Shlomo Shamai (Shitz) ‡ ∗ Departmen t of Electrical Eng ineering, Prin ceton University , Pr inceton, NJ 085 44, USA, { orens, poor } @prin ceton.edu † CWCSPR, Department o f Elec trical an d Com puter En gineering , NJIT , Ne wark, NJ 07102 , USA, o svaldo.simeone@njit.edu ‡ Departmen t of Electrical Eng ineering, T echnion, Haifa 32 000, Israel, sshlomo @ee.technion .ac.il Abstract — In this paper the advantages pro vided by mul- ticell processing of signals transmitted by mobile terminals (MTs) which ar e recei ved via dedicated relay terminals (R Ts) are studied. It is assumed th at each R T is capable of full- duplex operation and receiv es the transmission of adjacent relay term inals. Focusing on intra-cell TDMA and non-fading channels, a simplifi ed r elay-aided uplink cellular model based on a model intro duced by W yner is consid ered. Assuming a nomadic application in whi ch the RTs ar e oblivious to the M Ts’ codebooks, a f orm of distributed compress-and-forwa rd (CF) scheme with d ecoder side infor mation is employ ed. Th e per -cell sum-rate of the CF scheme is derived and is given as a solution of a simple fixed point equ ation. This achieva ble rate rev eals that the CF scheme is able to completely eliminate the in ter -r elay interference, and it approaches a “cut-set-like” upp er bound for strong RTs transmission power . The CF rate is also shown to surpass the rate of an amplify-and-forward scheme via numerical calculations for a wide range of the system parameters. I . I N T RO D U C T I O N T echn iques for providing high data rate services and better coverage in ce llular m obile commun ications are c urrently being investigated by many research g roups. In this pap er , we stu dy the com bination of two cooperatio n-based tec hnolo- gies that are p romising cand idates f or ach ieving such goals, extending p revious work in [1] - [4 ]. The first is relayin g, whereby the signal transmitted by a mob ile terminal (MT) is forwarded by a dedicated relay te rminal (R T) to the intended base station (BS) [ 5]. The secon d tech nology o f interest her e is m ulticell proce ssing (MCP), which allows the BSs to jointly decode the received signals, equ iv alently creating a distributed receiving an tenna arr ay [6]. The perfor mance gain provided by this tech nology within a simp lified cellular mod el was fir st studied in [7] , under the assumption that BSs are connected by an ideal back bone (see [8] for a survey on MCP). Recently , the interplay b etween these two tech nologies has been investigated for amp lify-and -forward (AF) a nd decod e- and-fo rward (DF) protocols in [1][4] and [2][ 3], respectiv ely . The basic f ramework employed in these works is the W yner uplink cellular model i ntrodu ced in [7]. According to the linear variant of this mod el, cells ar e arranged in a linear geometry and on ly adjacent cells interfere with each oth er . Moreover, inter-cell inter ference is described by a single par ameter α ∈ [0 , 1 ] , which defin es th e gain experien ced b y sign als trav e lling to interfered cells. Notwithstanding its simplicity , th is model Cell  m Cell  m-1 BSs: Cell  m+1 RTs: MTs: CP Backhaul: 1 s t  L a g 2 n d  L a g P S f r a g r e p l a c e m e n t s α α α α β β β γ γ γ µ µ µ µ η η η η Fig. 1. Schemati c diagram. captures the essential stru cture of a cellular system an d it provides insigh t into the sy stem performan ce. In this work we adopt a similar setup to the one presented in [4], in wh ich dedicated fu ll-duplex (FD) R Ts are added to th e ba sic lin ear W yn er u plink c hannel m odel and the signal path between adjacent R T s is consider ed (i.e., inter -r elay interferen ce). W ith coverage extension in mind, we focu s on distant users ha vin g no direct connec tion to the BSs. Assuming a nomadic a pplication in which the R Ts are oblivious to the MTs’ codeboo ks, a form of distributed compr ess-and-fo rward (CF) schem e with decode r side in formation , similar to that of [11], is a nalyzed. It is noted that this sch eme resemb les the si ngle-user multiple-relay CF scheme co nsidered in [9, Thm. 3]. Focusing on a setu p with infinitely large numbe r of cells, the achiev ab le p er-cell sum-r ate of the CF scheme is derived using the method s applied in [10]. The achievable rate, which is gi ven as a solu tion of a simple fixed p oint equ ation, shows that the CF scheme completely eliminates the inter- relay inter ferences. Moreover the rate is shown to app roach a “cut-set-like” u pper bound for strong R Ts transmission power . Finally , the per formanc e of the CF scheme is co mpared numerically with the p erform ance of an AF scheme, recently reported in [4], revealing th e superiority of th e CF scheme for a wide ran ge o f the system parameters. I I . S Y S T E M M O D E L W e c onsider the up link of a cellu lar system with a dedicated R T for each tr ansmitting MT . W e focus on a scenar io with no fading and ad opt a circu lar version of the linear cellular up link channel presented by W yner [7]. R Ts are add ed to th e basic W yner model fo llowing the analysis in [4] (see Fig. 1 fo r a schematic diag ram of a single ce ll within the setup and its inter-cell interaction ). The system includes M iden tical cells arr anged on a circle, with a single MT active in eac h ce ll at a given time (intr a- cell TDMA protoco l), and a dedicated single R T to relay the signals from the MT to the BS ( there is no dir ect con nection between MTs a nd BSs). Ac cording ly , each R T receives the signals o f the lo cal MT , the two adjacent MTs, and the two ad jacent R Ts, with ch annel power g ains β 2 , α 2 , and µ 2 respectively . Likewis e, each BS rec ei ves the sig nals of the local R T , and the two adjacent R Ts, with channel power gains η 2 and γ 2 respectively . The r eceiv e d signals at the R Ts and BSs are affected by i.i.d. zero-mean complex Gaussian additi ve noise processes with powers σ 2 1 and σ 2 2 , respecti vely . It is assumed that th e MTs use indep endent r andomly gener ated complex Gaussian co deboo ks with zero-m ean and power P , whereas the R T s are subjected to an average tran smit power constraint Q . Th e R Ts are assum ed to be o blivious of th e MTs codebo oks (nomad ic a pplication), an d that no coop eration among MTs is allowed. In add ition, the R Ts ar e assumed to be capa ble o f recei ving an d transmitting simultaneo usly (i.e., perfect echo-can cellation). It is noted that the propag ation delays between the different nodes of the system a re negligible with re spect to the symb ol du ration. Fin ally , it is assumed that the BSs are connec ted to a centr al processor (CP) via an ideal backhau l network, and th at the channe l path gain s and noise powers are known to the BSs, MTs, and CP . I I I . P R E L I M I N A R I E S A. W yner’s Model - Sum- Rate Capac ities Putting aside th e inter-relay interferen ce paths a nd the lac k of joint M CP among th e R Ts, the mesh n etwork of Fig. 1 is composed o f two W yn er mod els (or two “ W yner lags”) [7]. The close r elations of th e curren t setup an d the W y ner mo del renders th e following definitions usef ul in the sequel. The per-cell sum-rate cap acity of the linea r (or circ ular) W yner uplink chann el with infinitely large nu mber of ce lls ( M → ∞ ), no u ser co operation , optimal MCP , signal- to-noise ratio ( SNR) ρ , inter-cell inter ference factor a (e.g . α or η in Fig. 1 ), an d local path g ain b (e.g. β or γ in Fig. 1), is gi ven by [7] R w ( a, b, ρ ) = Z 1 0 log 2 ` 1 + ρH ( f ) 2 ´ d f . (1) where H ( f ) = b + 2 a cos 2 π f . Wh en tr ansmitter fu ll coo p- eration is allowed the per-cell su m rate capacity of the above channel is achieved by “waterfilling” solutio n and is given by R wf w ( a, b, ρ ) = Z 1 0 log 2 1 + „ ν − 1 H ( f ) 2 « + H ( f ) 2 ! d f s . t . Z 1 0 „ ν − 1 H ( f ) 2 « + = ρ , (2) where ( x ) + = min { x, 0 } . B. U pper B ound Denoting ρ 1 = P /σ 2 1 and ρ 2 = Q/σ 2 2 as the SNRs over the first “MT -R T ” and second “R T - BS” lags, resp ectiv ely , we have th e fo llowing bound. Proposition 1 The p er-cell sum- rate of a ny scheme employed in th e r elay-aided W yner cir cular u plink channel with infin ite number o f cells M → ∞ and no MT co operation, is u pper bound ed by R ub = min n R w ( α, β , ρ 1 ) , R wf w ( η , γ , ρ 2 ) o . (3) Pr oof: (ou tline) T he rate expression is easily derived by considerin g two cu t-sets, one separatin g the MTs from th e R Ts and the oth er separating the R Ts from th e BSs (or CP). W e refer to this b ound as “cut-set-like” bound since we also account fo r the assumptio n of n o MTs coop eration in the first lag. It is n oted that the u pper bou nd continues to hold even if we allow multiple MTs to be simultaneou sly ac ti ve in each cell (a ssuming a total-cell tran smit p ower of P ). Since both arguments of ( 3) increase with SNR it is easily verified th at R ub → ρ 1 →∞ R wf w ( η , γ , ρ 2 ) a nd that R ub → ρ 2 →∞ R w ( α, β , ρ 1 ) . C. Amp lify-and-F orwar d Scheme As a reference result, we consider the AF scheme with MCP ana lyzed in [4] fo r a similar in finite setup. For the AF scheme we make an add itional assumption regarding the relaying delay , nam ely that the R Ts amplify and forward the received signals with a delay o f λ ≥ 1 symbols (an integer). Interpr eting the cellular up link chan nel m odel with AF and MCP as a 2D linear time in variant system, and applying the 2D extension of Szeg ¨ o’ s theorem [7], the following resu lt is derived in [4]. Proposition 2 An a chievable per -cell sum-rate of AF relaying with op timal MCP an d no spe ctral shaping, employed in the r elay- aided infinite linear W yner uplink cha nnel, is given by R af = Z 1 0 log A + B + p ( A + B ) 2 − C 2 B + √ B 2 − C 2 ! d f , (4) wher e A , P g 2 ( β + 2 α cos 2 π f ) 2 ( γ + 2 η cos 2 π f ) 2 B , σ 2 1 g 2 ( γ + 2 η cos 2 π f ) 2 + σ 2 2 (1 + 4 g 2 µ 2 cos 2 2 π f ) C , 4 σ 2 2 g µ cos 2 π f . Furthermore , the o ptimal r elay gain g o is the u nique solution to th e equation σ 2 r ( g ) = Q wher e σ 2 r ( g ) = ( P β 2 + σ 2 1 ) g 2 p 1 − (2 µg ) 4 + 4 P α 2 g 2 p 1 − (2 µg ) 2 + 1 − (2 µg ) 2 (5) ) ( ) 1 ( 1 , ) 1 ( 1 , m m W X ) ( ) 1 ( 3 , ) 1 ( 3 , m m W X ) ( ) 1 ( 2 , ) 1 ( 2 , m m W X ) 1 ( 2 , m Y ) 1 ( 1 , m Y ) 1 ( 3 , m Y ) 2 ( 2 , ) 1 ( 1 , m m W Y → Block 1 Block 3 Block 2 Transmit Receive Compress Transmit Receive Decode Decompress and Decode ) 2 ( 3 , ) 1 ( 2 , m m W Y → ) ( ) 2 ( 2 , ) 2 ( 2 , m m W X ) ( ) 2 ( 2 , ) 2 ( 3 , m m W X CP BS RT MT ) 2 ( 2 , m Y ) 2 ( 3 , m Y } , { } { 2 , ) 2 ( 2 , ) 2 ( 2 , m m m T W Y → } , { } { 3 , ) 2 ( 3 , ) 2 ( 3 , m m m T W Y → } { } { ) 1 ( 1 , ) 2 ( 2 , m m W W → } { } , { ) 1 ( 2 , 2 , ) 2 ( 3 , m m m W T W → Fig. 2. CF s cheme diagram. is th e r elay outpu t power . It is shown in [4] th at the optimal gain is achiev ed wh en the relays use th eir f ull power Q , and that g o − → Q →∞ 1 / (2 µ ) . In addition, R af is not in terferenc e limited and it is inde pendent of the ac tual R T delay value λ . I V . D I S T R I BU T E D C O M P R E S S - A N D - F O RW A R D S C H E M E Here we describe the p roposed CF-based transmission scheme, which organiz es tr ansmission in to succ essi ve blo cks (or cod ew or ds) o f N symbo ls, as sketched in Fig . 2. It shou ld be remar ked th at tran smission in the AF scheme presented in the previous section spans only one b lock (with some o ( N ) symbols margin due to the delay λ an d the filter e ffecti ve response time). For this r eason, while in the AF schem e the R Ts need to maintain on ly symbo l synchro nization, for the CF scheme to be discussed below , block synchr onization is also necessary . W ith ( · ) (1) , ( · ) (2) denoting the a ssociation to the first “MT - R T” and second “R T -BS” lags, respectively , the received sign al at the m th R T in a n arb itrary sym bol of the n th block is Y (1) m,n = β X (1) m,n + α ( X (1) [ m − 1] ,n + X (1) [ m +1] ,n ) + T m,n + Z (1) m,n , (6) where [ k ] , k mod M , X (1) m,n are the signals transmitted by the MTs (to be de fined in the sequ el), Z (1) m,n denotes the additive no ise a t th e R T , and the inter -relay inte rference is T m,n = µ ( X (2) [ m − 1] ,n + X (2) [ m +1] ,n ) . (7) The recei ved signal at the m th BS is Y (2) m,n = γ X (2) m,n + η ( X (2) [ m − 1] ,n + X (2) [ m +1] ,n ) + Z (2) m,n , (8) where X (2) m,n are the sign als tr ansmitted by th e R Ts, and Z (2) m,n denotes the add itiv e noise at the BS. The proposed C F scheme works as follows (see Fig. 2 where shadowed boxes indicate zero time pro cessing). The basic idea is to have the R T s comp ress the signal Y (1) m,n received in any n th block (say n = 2 in Fig. 2) and forward it in th e ( n + 1) th block ( e.g., n + 1 = 3) via a channel c odeword X (2) m,n +1 , by exploiting the side in formatio n available at the CP abou t the compressed signals Y (1) m,n . In fact, with the propo sed scheme, in the n th b lock, the CP d ecodes the ch annel co dew o rds X (2) m,n transmitted by th e R Ts, and these are co rrelated with the signal Y (1) m,n (6) via T m,n (7). Based on this side info rmation, distributed CF is implemented at the R Ts accordin g to [11] via standard vector qu antization and binn ing. A more formal description of the CF scheme is p resented b elow . Code Construction : 1) At the MTs : each m th MT generates a rate- R cf Gaussian r andom chan nel co deboo k X (1) m accordin g to C N (0 , ρ 1 ) (no op timality is claimed); 2) At the RTs : 2.a) Each R T gener ates a ra te- R w ( η , γ , ρ 2 ) Gaussian rand om channel cod ebook X (2) m accordin g to C N (0 , ρ 2 ) ; 2.b) Each R T gener ates a rate- ˆ R = I ( Y (1) m ; U m ) Gaussian quantizatio n codebo ok U m accordin g to the margina l distribution induced by U m = Y (1) m + V m , (9) where the quantiza tion no ises V m are i.i.d. zero-m ean com - plex Gaussian in depend ent of all other rando m variables (no o ptimality is claimed). Each quantizatio n codebo ok is random ly partitio ned into 2 N R w ( η, γ ,ρ 2 ) bins, each of size 2 N ( ˆ R − R w ( η, γ ,ρ 2 )) ; Encoding at the MTs : each MT sends its message W (1) m,n ∈ W (1) = { 1 , . . . , 2 N R cf } b y tran smitting the N sym- bol vector X (1) m,n = X (1) m ( W (1) m,n ) over the first “MT - R T” lag; Processing a t the RTs : 1) Comp r essing : each R T employs vector quantizatio n usin g standard joint typicality arguments via the quan tization cod ebok U m , to compress the pr eviou sly received vector Y (1) m,n − 1 into U m,n with the correspo nding bin ind ex W (2) m,n ; 2 ) E ncoding : each R T sends its bin in - dex W (2) m,n ∈ W (2) = { 1 , . . . , 2 N R w ( η, γ ,ρ 2 ) } b y tran smitting X (2) m,n = X (2) m ( W (2) m,n ) over the second “R T -BS” lag; Decoding at the CP : 1) Deco ding the bin indices : the CP collects the rece i ved signal vectors Y (2) M ,n (where M = { 0 , 1 , . . . , M − 1 } ) from a ll the BSs thro ugh th e back- haul link s. T hen it d ecodes the resulting multiple-acce ss cha n- nel ( MA C) u sing stand ard method s (e.g., [1 2]) to recover an estimate ˆ W (2) M ,n ; 2) Comp osing the side in formation : the CP uses the d ecoded bin ind ices ˆ W (2) M ,n to co mpose the side info r- mation vector s ˆ T M ,n , whe re ˆ T m,n = µ ( ˆ X (2) [ m − 1] ,n + ˆ X (2) [ m +1] ,n ) , to be used in the next b lock; 3) Decoding the MTs messages : The CP uses the p r evious sid e informatio n ˆ T M ,n − 1 and looks for a unique joint typical triplet { X (1) M ,n − 1 , U M ,n , ˆ T M ,n − 1 } within the bins ind icated by ˆ W (2) M ,n , acc ording to the joint distribution induced by ( 6), to recover ˆ W (1) M ,n − 1 . V . S U M - R AT E A NA LY S I S Here we derive th e per-cell su m-rate (o r sym metric rate) achiev ab le via the p roposed CF scheme . Proposition 3 An achievable p er-cell su m-rate of the CF scheme employed in th e r elay-a ided W yn er cir cular uplink channel with infinite number o f cells M → ∞ , is given by R cf = R w “ α, β , ρ 1 (1 − 2 − r ∗ ) ” , (10) wher e r ∗ ≥ 0 is the u nique solution to the following fixed point equation R w “ α, β , ρ 1 (1 − 2 − r ∗ ) ” = R w ( η , γ , ρ 2 ) − r ∗ . (11) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 Per−Cell Sum−Rates ( σ 2 1 = σ 2 2 =0 [dB] , P=10 [dB], Q=20, α = η =0.2, β = γ =1) Inter−Relay Interference Factor µ R [bits/channel use] Upper Bound CF MCP AF MCP Fig. 3. Rates vs. the inter-rela y interference facto r µ (symmetrical hops). Pr oof: (outline) See App endix A. It is con cluded tha t th e rate R cf is inde penden t of the inter- relay interferen ce. Moreover , the CF scheme per forms as if there are no inter-relay inter ferences (i.e . µ = 0 ) and its rate coincides with th e resu lts o f [10] interp reting th e second “R T -BS” lag as the backhau l n etwork with limited capa city C = R w ( η , γ , ρ 2 ) . Also n ote, that the resu lt hold s even if we relax the R T pe rfect echo-can cellation assump tion a s long as the CP is aware of the residual ech o p ower gain. Since R w is giv en in an implicit in tegral fo rm ( 1), we can not solve the fixed po int equation (11) an alytically . Never- theless, since R w ( α, β , ρ 1 (1 − 2 − r )) is mono tonic in r , (11) is easily so lved numerically . It is also evident that the CF rate increases with the relay power Q . Hence, as with the AF scheme full relay power usage is optimal. It is easily verified th at whe n ρ 1 → ∞ then r ∗ → 0 , and R cf does not ach iev e th e up per bou nd (3). This is since R cf → ρ 1 →∞ R w ( η , γ , ρ 2 ) ≤ R wf w ( η , γ , ρ 2 ) . On the other extreme when ρ 2 → ∞ th en r ∗ → ∞ , and R cf achieves th e upper bound R cf → ρ 2 →∞ R w ( α, β , ρ 1 ) . I n the next sectio n, nu merical results r ev e al that the CF schem e o utperfo rms th e AF scheme over a wide rang e of the system p arameters. V I . N U M E R I C A L R E S U LT S In Fig. 3 the per-cell sum-rates of the CF and AF schemes are plotted alo ng with the upp er bou nd (3), as fu nctions of the in ter-relay interferen ce factor µ f or ρ 1 = P /σ 2 = 1 0 [dB], ρ 2 = Q / σ 2 = 20 [dB], σ 2 1 = σ 2 2 = σ 2 = 1 , α = η = 0 . 2 , and β = γ = 1 . I t is noted tha t the AF cu rve is p lotted with optimal relay gain (resu lting in a f ull usag e of the relay power Q ). Examining the figure, the ben efits of th e CF scheme are evident in view of the deleteriou s effect of in creasing in ter- relay interferen ce µ on the AF rate; the CF sch eme provides almost twic e the bits per chann el use than the AF rate, fo r strong in ter-relay interfer ence lev els. Also visible from the figure is the proximity of th e CF rate to the up per bo und (less than 0.2 bits per ch annel use) which for this setting is dominated by th e rate of the fir st “MT -R T” lag. 0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 Per−Cell Sum−Rates ( σ 2 1 = σ 2 2 =0 [dB], Q=20 [dB] , α = η =0.2, β = γ =1) MT Transmit Power P [dB] R [bits/channel use] Upper Bound CF MCP AF MCP ( µ =0) AF MCP ( µ =0.8) Fig. 4. Rates vs. the MT s transmission powe r P (symmetrical hops). 0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 7 Per−Cell Sum−Rates ( σ 2 1 = σ 2 2 =1 [dB], Q=20 [dB] , α =0.6, β =1, η =0.2, γ =1) MT Transmit Power P [dB] R [bits/channel use] Upper Bound CF MCP AF MCP ( µ =0) AF MCP ( µ =0.8) Fig. 5. Rates vs. the MT s transmission power P (asymmetrical hops). Figures 4 and 5 presen t the CF an d AF rate cur ves an d the upper-boun d as fun ctions of the MTs power P , for ρ 2 = Q / σ 2 = 20 [dB], σ 2 1 = σ 2 2 = 1 , and β = γ = 1 . Here, we f ocus on two scenarios: (a) a setting with sym metrical first and seco nd lags, i.e. α = η = 0 . 2 , a nd (b) a setting with asymmetrical lags, i.e. α = 0 . 6 , η = 0 . 2 . In bo th figu res we inclu de the AF rate curves fo r the two extrem es µ = 0 and µ = 0 . 8 , wh ich repr esent weak and stro ng in ter-relay interferen ce scenarios, respectively . It is noted that any AF rate curve with 0 < µ < 0 . 8 is confined between these two curves. Examinin g the fig ures it is observed th at the CF p erforms well (within on e bit pe r channel use o f the up per boun d) in both scenarios over the entire displayed ra nge of MTs power P . On the oth er hand, the AF scheme per forms well in both scenarios only for low inter-relay interference le vels and low MTs power P . Other re sults (not presente d h ere) show that the AF scheme is slightly ben eficial over th e CF sch eme under certain cond itions ( e.g., h igh P , and asymmetr ical lags α = 0 . 2 , η = 0 . 6 ). V I I . C O N C L U D I N G R E M A R K S In this work we h av e con sidered a simp lified two-ho p cellu- lar setup with FD relays an d in ter-relay in terference. Focusing on no madic app lication, a f orm of distributed CF with d ecoder side info rmation scheme, has been analyzed. W e have d erived the ach ie vable per-cell sum -rate for an infinitely large number of cells, and have shown that the CF sche me totally elimina tes the inter-relay interfe rence. Numerical results reveal that the CF ra te curves are ra ther close to a “cut-set-like” up per bound , and also demon strate the sup eriority o f the propo sed CF scheme over the MCP AF scheme o f [4], for a wide r ange of the system parame ters. A P P E N D I X A. P r oof of P r oposition 2 ( Outline) W e fo cus on the d ecoding stage at th e CP for th e n th block (recall Fig . 2) . Since the rate of th e ch annel codebo oks used by the R Ts on the second lag is eq ual to the per-cell capacity R w of the corr esponding W yner chann el (see Sec. III-A), the CP is able to cor rectly d ecode W (2) M ,n − 1 from the previous b lock and W (2) M ,n from the current with high probab ility . Based on the f ormer, it can also build an accurate estimate ˆ T M ,n − 1 . As per Fig. 2, the CP the n attempts to decode the m essages W (1) M ,n − 1 . In the following, th e variables of interest are Y (1) m,n − 1 , U m,n and X (1) m,n − 1 which are denoted for simplicity as Y m , U m and X m . T o elabo rate, we n ote that, due to the quan tization ru le (9), the following Markov rela tion holds { X M , U M\ m , T M } − Y m − U m . Recall also that the CP deco des X M by lo oking for jointly ty pical sequ ences { X M , U M , ˆ T M } , where X M belong to the MTs codebook s (each of size 2 N R cf ) and U M are within the b ins (of size 2 N ( ˆ R − R w ( η, γ ,ρ 2 )) ) whose in dices are given by W (2) M ,n . Assuming ˆ R ≥ I ( Y m ; U m ) , fo r large block leng th N , the probab ility o f error is domin ated b y the events wh ere a set with erroneo us X L and U S , for any sub sets L , S ⊆ M , is fou nd that is jointly typical in the sense explained ab ove (see [10]). Using the u nion boun d, we found that the error pro bability is bound ed P e ≤ X L , S ⊆M 2 N R cf |L| + N ( ˆ R − R w ) |S | · 2 N ( h ( X L ,U S | X L c ,U S c ,T M ) −|L| h ( X ) −|S | h ( U )) . It fo llows tha t, in o rder to driv e the probability of err or to zero, it is sufficient that |L| R cf + |S | ( ˆ R − R w ) ≤ − h ( X L , U S | X L c , U S c , T M ) + |L| h ( X m ) + |S | h ( U m ) . (12) Now , defining ˜ Y m = Y m − T m and ˜ U m = ˜ Y (1) m + V m , an d using the Markov pr operties of the compr ession scheme, we have th at I ( Y m ; U m ) = h ( U m ) − h ( U m | Y m ) = h ( U m ) − h ( U m | Y m , X M , T M ) = h ( U m ) − h ( ˜ U m | ˜ Y m , X M ) , (13) and h ( X L , U S | X L c , U S c , T M ) = = h ( X L | X L c , U S c , T M ) + h ( U S | X M , U S c , T M ) = h ( X L | X L c , U S c , T M ) + |S | h ( U m | X M , T M ) = h ( X L | X L c , ˜ U S c ) + |S | h ( ˜ U m | X M ) , (14) and it is also easy to prove that |L| h ( X m ) = h ( X L ) = h ( X L | X L c ) . (15) Using ( 13)-(15) in ( 12) and dr opping the sub script d enoting the ce ll in dex fo r symmetry , we g et |L| R cf ≤ |S | ( R w − I ( ˜ U ; ˜ Y | X M )) + I ( X L ; ˜ U S c | X L c ) , which corre sponds to the result in [10] b y substitution of ˜ U and ˜ Y with U and Y , an d th e pro of is comp leted by f ollowing [10]. A C K N O W L E D G M E N T The resear ch was supported by a Marie Curie Outgo ing Internatio nal Fello wsh ip and th e NEWCOM++ ne twork of excellence both within the 6th and 7 th European Com munity Framework Prog rammes, by th e U.S. National Science Foun- dation und er Gran ts AN I-03-3 8807 and CNS-06- 25637 , and the REMON con sortium fo r wire less co mmunicatio n. R E F E R E N C E S [1] O. Simeone, O. Somekh, Y . Bar-Ness, and U. Spagnolini, “Uplink throughput of TDMA cellu lar systems with multice ll processing and amplify-a nd-forwa rd coopera tion between mobiles, ” IEE E T rans. W ir e- less Commun. , pp. 2942–2951, Aug. 2007. [2] O. Simeone , O. Somekh, Y . Bar-Ness, and U. Spagnolini, “Throughput of low-po wer cel lular systems with collaborati ve base stations and relayi ng, ” IEEE T rans. on Inform. Theory , vol. 54, pp. 459-467, Jan. 2008. [3] O. Si meone, O. Somekh, Y . Bar-Ness, H. V . Poor, and S. Shamai (Shitz), “Capac ity of linear two-hop m esh netw orks with rate splitting, decode- and-forw ard relaying and cooperation, ” in Proc . of the 45th Annual Allerton Confere nce on Communication , Contr ol and Computing , (Mon- ticel lo, IL), Sep. 26–28 2007. [4] O. Som ekh, O. Simeone, H. V . Poor , and S. Shamai (Shitz), “Cellular systems with full-duple x amplify-and-forw ard relaying and coopera tive base-stat ions, ” in Pr oc. of the IEE E Intl. Symp. on Inform. Theory (ISIT) , (Nice, France), pp. 16–20, Jun. 2007. [5] Y . -D. J. L in and Y .-C. Hsu, “Multihop cellu lar: A new architectu re for wireless communicati ons, ” in Pro c. of the IEEE INFOCOM (3) , (T el- A viv , Israel), pp. 1273–1282, Mar . 26–30, 2000. [6] S. Zhou, M. Zhao, X. Xu, and Y . Y ao, “Distri bute d wireless commu- nicat ion system: a ne w archi tectu re for public wireless access, ” IEE E Commun. Magazine , pp. 108–113, Mar . 2003. [7] A. D. W yner , “Shannon-theore tic approach to a Gaussian cellula r multiple -access channel, ” IEEE Tr ans. on Inform. Theory , vol. 40, pp. 1713–1727, Nov . 1994. [8] O. Somekh, O. Simeone, Y . Bar -Ness, A. M. Ha imovich, U. Spagnolini, and S. Shamai (Sh itz), Distrib uted A ntenna Syst ems: Open Arc hitectur e for Future W irele ss Communic ations , ch. An Information Theoretic V ie w of Distrib uted Antenna Processing in Cellu lar Systems. Auerbach Publica tions, CRC Press, May 2007. [9] G. Kramer , M. Gastpar , and P . Gupta, “Coopera ti ve strate gies and capac ity theorems for relay networks, ” IEE E T rans. on Inform. Theory , vol. 51, pp. 3037– 3063, Sep. 2005. [10] A. Sanderovi ch, O. Somekh, and S. Shamai (Shitz), “Uplink macro di versity with limited backhau l capac ity , ” in Pr oc. of the IEEE Intl. Symp. on Inform. T heory (ISIT) , (Nice, France), pp. 11–15, Jun. 2007. [11] M. Gastpar , “The W yner-Ziv problem with multiple s ources, ” IEE E T rans. on Inform. Theory , vol. 50, pp. 2762–2768, Nov . 2004. [12] T . M. Cove r and J. A. Thomas, Elements of Information Theory . J ohn W iley and Sons, Inc., 1991.