On the Capacity of Cloud Radio Access Networks

Transcription

1 07 IEEE Inernaional Symposium on Informaion Theory ISIT On he Capaiy of Cloud Radio Aess Neworks Shouvik Ganguly and Young-Han Kim Deparmen of Elerial and Compuer Engineering Universiy of California, San Diego Absra Uplink and downlink loud radio aess neworks are modeled as wo-hop K-user L-relay neworks, whereby small base-saions a as relays and are onneed o a enral proessor via orhogonal links of finie apaiy Simplified versions of noisy nework oding and disribued deode forward are used o esablish inner bounds on he apaiy region for uplink and downlink ommuniaions, respeively Through a areful analysis, he uplink inner bound is shown o ahieve he use bound on he apaiy region universally wihin O L bis per user The downlink inner bound ahieves he use bound wih a slighly looser gap of OKL These igh per-user gap resuls are exended o he siuaions in whih he nodes have muliple anennas I INTRODUCTION Wih ever-inreasing demands for higher daa raes, beer overage, and reliabiliy of ommuniaion for a large number of devies, novel nework proools and arhieures are expeed o play an imporan role in fuure ommuniaion sysems The loud radio aess nework C-RAN arhieure ] is one of he promising andidaes, in whih ommuniaion over a group of ells is oordinaed by a loud-based enral proessor Fig depis C-RAN uplink and downlink sysems shemaially Base-saions in he C-RAN arhieure, unlike radiional wireless sysems, do no perform he omplee proessing loally, bu are insead onneed o he enral proessor hrough wired or wireless fronhaul links If hese links have unbounded apaiies, he C-RAN an be viewed as a disribued muliple inpu muliple oupu MIMO sysem Base-saions a as remoe radio heads ha use beamforming o oordinae ransmission and miigae inerferene among muliple ells For he more realisi siuaion of limied apaiies, he opimal beamforming soluion is ypially ompued, assuming infinie fronhaul apaiies, and hen ompressed individually, whih is hen applied a he base-saions As an alernaive o his greedy beamforming-ompression approah, his paper invesigaes he opimal oding sheme for he enire sysem by modeling he C-RAN as a wo-hop relay nework In his model, he base-saions a as relays ha summarize he reeived signals o he enral proessor uplink and ransmi he presribed signals from he enral proessor downlink ] The wo-hop relay nework model for he uplink C-RAN was sudied by Zhou e al ], who applied nework ompress forward 3] o his ase and showed, by opimizing over quanizers, ha under some symmery assumpions, i is possible o ahieve a sum rae gap from he use bound ha is Cenral proessor Radio heads User devies Fig : a Uplink and b downlink loud radio aess nework linear in he number of base-saions Sanderovih e al 4] used he same sheme wihou opimizaion and analyzed he large-user asympois of ahievable raes under he senario where eah fronhaul link has a fixed apaiy Zhou e al 5] subsequenly showed ha under a sum-apaiy onsrain on he fronhaul links, he oding sheme in 4] and ] an be simplified hrough suessive anellaion deoding, generalizing an earlier single-user muliple relay sheme 6] In his paper, we apply noisy nework oding 7] o he uplink wo-hop relay nework model and he sheme ahieves wihin O L bis per user from he apaiy region, where L is he number of relays base-saions, regardless of he hannel gain marix, power onsrain, and he number of users Compared o nework ompress forward, noisy nework oding is simpler in ha relays send he ompression indies of he reeived signals direly wihou binning hashing This simpler operaion iner alia ahieves higher raes han nework ompress forward for general neworks, bu for he uplink C-RAN, he ahievable rae regions oinide Hene, our main onribuion for he uplink C-RAN an be viewed as a refinemen of he apaiy analyses in 4] and ] For he downlink, a variey of oding shemes have been proposed Hong and Caire 8] sudied a low-omplexiy reverse ompue forward sheme for symmeri raes Liu e al 9] applied nework oding and beamforming o he downlink model wih a noiseless muli-hop fronhaul Moivaed by he MAC BC dualiy, Liu e al 0] proposed subopimal ompression-based shemes and esablished a dualiy beween ahievable rae regions for he uplink and downlink C-RANs In his paper, as an alernaive approah, we speialize /7/$ IEEE 063

2 07 IEEE Inernaional Symposium on Informaion Theory ISIT and simplify he disribued deode forward oding sheme ] o he downlink C-RAN wih apaiy-limied single-hop fronhaul In his sheme, mulioding a he enoder as in Maron oding for broadas hannels ] is oupled wih oding for fronhaul links, whih allows more effiien oordinaion among he ransmied signals a he base-saions We show ha our rae region ahieves a per-user gap of 45+LK from he use bound, where L and K are he number of relays and users This refines he bes-known linear gap from apaiy for his model implii in 0] The res of he paper is organized as follows Seion II sudies he uplink model Seion II-A desribes he general inner and ouer bounds on he apaiy region; Seion II-B speializes he noisy nework oding inner bound o he Gaussian nework model and esblishes he apaiy gap; and Seion II-C generalizes he gap resul o he MIMO ase Seion III parallels he same flow for he downlink C-RAN Throughou he paper, we follow he noaion in ] In addiion, A F := raa T = ra T A denoes he Frobenius norm of a marix A All arihms are o base and all informaion measures are in bis A General Model II UPLINK COMMUNICATION We model he uplink C-RAN as a wo-hop relay nework in Fig, where he firs hop, namely, he wireless hannel from he user devies o he radio heads, is modeled as a disree memoryless nework py L x K, and he seond hop, namely, he hannel from he radio heads o he enral proessor, onsiss of orhogonal links of apaiies C,, C L bis per ransmission, deoupled from he firs hop To be more preise, he hannel oupu a he enral proessor reeiver is W,, W L, where W l : nc l ] is a reliable esimae of wha relay l ommuniaes o he reeiver over n ransmissions We assume wihou loss of generaliy ha hese ommuniaion links are noiseless X XK py L x K Y Y L Fig : Uplink nework model A nr,, nr K, n ode for his nework onsiss of K message ses : nr ],, : nr K ]; K enoders, where enoder k : K] assigns a odeword x n k o eah m k : nr k ]; L relay enoders, where relay enoder l : L] assigns an index w l : nc l ] o eah reeived sequene yl n ; and a deoder ha assigns message esimaes ˆm,, ˆm K o eah index uple w L We assume ha he messages M,, M K are uniformly disribued and independen of eah oher The average probabiliy of error is defined as P e n = P K k= ˆM k M k } A rae uple R,, R K is ahievable if here is a sequene of nr,, nr K, n C C L Y odes wih lim n P e n = 0 The apaiy region is defined as he losure of he se of all ahievable rae uples We have he following inner bound 4] on he apaiy region of his nework Proposiion : A rae uple R,, R K is ahievable if R k < IXS ; Ŷ S XS k S + l S C l l S IY l ; Ŷl X K for all S : K] and S : L] for some pmf K k= px k L l= pŷ l y l This inner bound was esablished by speializing he nework ompress forward sheme in 6] Roughly speaking, eah relay ompresses is reeived sequene and sends he bin index of he ompression index A more sraighforward sheme an be developed by speializing noisy nework oding 7] and simplifying i o our nework model This simplified oding sheme and is analysis will be presened in a longer version of his manusrip and is omied here The use bound 3] for his nework an be haraerized by he se of R,, R K suh ha R k < IXS ; Y S XS + C l k S l S for all S : K] and S : L] for some pmf px K B Gaussian Model We now assume ha Y L = GX K + Z L, where G R L K is a deerminisi hannel gain marix and Z L is a veor of independen N0, noise omponens Assume he average power onsrain P on eah sender, ie, n x kim k np, m k : nr k ], k : K] i= Our main goal of his seion is o quanify how well nework ompress forward or noisy nework oding, wih ahievable raes in, performs for his Gaussian nework In pariular, we bound he per-user rae gap suh ha if R,, R K lies in he use bound, he rae uple R,, R K will lie in he inner bound in, regardless of G and P I an be shown ha Proposiion an be simplified o esablish he ahievabiliy of all R,, R K suh ha R k < P + G S G T,S S + I,S k S + C l S + l S =: f in S, S, 3 where G S,S is he submarix of G formed by he rows wih indies in S and he olumns wih indies in S This follows by onsidering X K o be a veor of iid N0, P random variables, and seing Ŷl = Y l + Ẑl, l : L], 064

3 07 IEEE Inernaional Symposium on Informaion Theory ISIT where Ẑl N0, Noe ha for any S : L] and S S, G S,S G T S G,S S,S GT S and hus,,s f in S, S f in S, S, whih implies ha min f in S S, S min f in S, S 4 S The use bound in an also be simplified and relaxed as R k < G S,S Σ S S GT S + I + C,S l k S l S a P GS,S G T S + I S +,S + C l l S =: f ou S, S, 5 where a follows in a similar manner as equaion 34 in Seion V of ], and Σ S S is he ondiional ovariane marix of XS given XS For his Gaussian nework model, we have he following upper bound on he gap from apaiy Theorem : For every G R L K and every P R +, if a rae uple R,, R K is in he use bound 5, hen he rae uple R +,, R K + is ahievable, where := Proof: Le S :K] S 45 + L min S f ou S, S min S f in S, S 6 S Suppose ha R,, R K lies in he use bound, and le A = k : R k > } Then, for every nonempy S : K], R k + k S = R k k S A = k S A R k S A a ] min f ou S A, S S = min S f in S A, S b min S f in S, S, min f ou S A, S min f in S A, S S S where a follows from he use bound 5, and he fa ha min S f ou S, S min S f in S, S = S S min S f ou S A, S min S f in S A, S, S A and b follows from 4 Hene,, as defined in 6 saisfies he requiremens of Theorem Now, for every > 0, min S f ou S, S min S f in S, S = S S a f ou S, S f in S, S S,S S b = S,S S P G S,S G T S + I,S P + S + G S,SGT S + I,S rankgs,s = S,S S i= + S P β i + P + β i + + S + ] d minl l, k} + k :K] k l 0:L] + l k + + ] 7 Here, a follows from he fa ha for funions f and g defined over a finie se X, suh ha g f everywhere on X, min x X gx min x X fx x X gx fx], b follows from 5 and 3, and in, β, β, are he non-negaive eigen-values of G S,S G T S,S Finally, in d, we ake S = k, S = l, and upper-bound rankg S,S by minl l, k} The imizaion in 7 yields + + L + +,, L + +, Sine his holds for every > 0, we se = L for L o obain + + L L + L L + L L ln L 8 For L =, we ake = o obain bi This, ogeher wih 8, esablishes Theorem C MIMO Model We now generalize Theorem o he siuaion in whih he senders and relays have muliple anennas For simpliiy, we assume ha every sender has anennas and every relay has r anennas We also assume he average power onsrain P a eah ransmi anenna Proposiion : If R,, R K lies in he use bound, hen R +,, R K + is ahievable, where 45 + Lr, Lr > 9 Lr+, Lr Proof skeh: Firs assume ha = In his ase, he sequene of seps leading o 3 and 5 go hrough almos 065

4 07 IEEE Inernaional Symposium on Informaion Theory ISIT unhanged, exep for a sligh hange of noaion in ha G S,S is now he r S S hannel gain marix from he senders in S o he relays in S Also, he las erm in 3 beomes r S + The relaion 7 now reads: minl lr, k} + k,l k + lr k + + ] Manipulaing his expression as before, we show ha 45 + Lr For general r and, 7 beomes minl lr, k} + k,l k + lr k + + ] For Lr, he imizaion, followed by subsiuing =, yields Lr+ For Lr > and, he imizaion yields + + Lr + + ; by seing = Lr, we have Lr Lr Lr/ rl Lr Lr + Lr/ + Lr Lr ln = 45 + Lr/, whih esablishes 9 A General Model III DOWNLINK COMMUNICATION Similar o he uplink, we model he downlink C-RAN as a wo-hop relay nework in Fig 3, where he firs hop enral proessor o radio heads onsiss of orhogonal noiseless links of apaiies C,, C L bis per ransmission and he seond hop radio heads o user devies is modeled as a disree memoryless nework py K x L A nr,, nr K, n ode for his nework onsiss of K message ses : nr ],, : nr K ]; an enoder w L m,, m K L l= : nc l ]; relay enoders x n l w l, l : L]; and deoders ˆm k yk n : nr k ], k : K] The average probabiliy of error, ahievabiliy of a rae uple, and he apaiy region are defined as before The following inner bound an be esablished by speializing he disribued deode forward sheme ] We desribe a simplified version of he oding sheme in he Appendix Proposiion 3: A rae uple R,, R K is ahievable, if R k < IXS ; US XS + k S IU k ; X L Y k k S l S for all S : L] and S : K] for some pmf L l= px l K k= pu k x L C l X C C L X X L py K x L Fig 3: Downlink nework model The use bound for his nework is haraerized by R k < IXS ; Y S XS + C l 0 k S l S for all S : L] and S : K] for some pmf px L B Gaussian Model Similar o Seion II-B, we now assume ha Y K = GX L + Z K, where G R K L is a hannel gain marix and Z K onsiss of iid N0, noise omponens, and assume he average power onsrain P a eah relay The res of his seion is devoed o bounding he ahievable per-user rae gap from he use bound Firs, Proposiion 3 an be simplified o esablish he ahievabiliy of all R,, R K suh ha k S R k < P G S,SGT S,S + I + l S Y Y K C l S + =: F in S, S for S : L] and S : K] This follows by seing X L o be a veor of iid N0, P random variables and defining U K = GX L + ẐK, where ẐK N0, I is independen of Z K The use bound 0 simplifies o R k < GS,S Σ S S GT S + I +,S k S l S =: F ou S, S for all S : L] and S : K] We now have he following Theorem : For every G and P, if R,, R K is in he use bound, hen R +,, R K + is ahievable, where 45 + KL Proof: Noe ha unlike 4, F in is no neessarily monooni We overome his diffiuly by rephrasing he inner bound as R k < min F in S, T 3 T S k S We observe ha he righ-hand side of 3 is inreasing wih S for a fixed S, so we an apply he ehnique developed in Seion II-B o ompue an upper bound on We hus wrie = S :K] min S F ou S, S S C l 066

5 07 IEEE Inernaional Symposium on Informaion Theory ISIT ] min S min T S F in S, T S F ou S, S F in S, T S :L] S S :K] T S = S :L] S :K] T S a S :L] S :K] T S S S G S,S Σ S S GT S,S + I P G T,S G T T,S + I + T + ] G S,S Σ S G T S,S + I P G S,S G T S,S + I + T G S Σ,S S G T S + I,S P G S,S G T S + I =,S + ], 4 where a follows sine Σ S Σ S S and for any marix A and α > 0, I + αaa T inreases when we add more rows o A Wriing Σ S = UΛU T, where U is orhogonal and Λ is diagonal, and leing G S,S U = b b b S ], where b,, b S are S veors onsrained by S l= b l = G S,S F, we have I + S a S b = l= λ lb l b T l l= b lb T l I + P S I + P S S l= b lb T l I + P S l= b lb T l + P S µ k + P k= µ k S S, provided ha S Here, a follows sine he rae of Σ S is upper bounded by P S and in b, µ,, µ S are he nonnegaive eigenvalues of S l= b lb T l Coninuing from 4, we hus have T + S + ] S S :L] S :K] T S = K + + L This holds for every, so we se = K o obain L + K K K K a L + K + ln a follows sine d/dxx x = x + / ln and he laer is an inreasing funion of x C MIMO Model As before, assume ha every relay has anennas and every reeiver has r anennas wih average power onsrain P a eah anenna By slighly modifying he argumens of he previous seion, i an be shown ha he following holds Proposiion 4: If R,, R K lies in he use bound, hen R +,, R K + is ahievable, where r L + K + 45, L > r L r + K + 45, L r APPENDIX The following oding sheme is a speializaion of disribued deode forward o our general downlink C-RAN model and ahieves he inner bound in Proposiion 3 Codebook generaion: Fix a pmf px L K k= pu k x L For eah w l, l : L], generae x n l w l n i= p X l x li Define auxiliary indies s k : n R k ], k : K] Here, eah R k is some non-negaive auxiliary rae For eah m k, s k : nr k ] : n R k ] and k : K], generae u n k m k, s k n i= p U k u ki Enoding: The enoder sends w L suh ha x n w,, x n Lw L, u n m, s,, u n Km K, s K T n ɛ Relay enoding: Relay l ransmis x n l w l Deoding: Le ɛ > ɛ Upon reeiving yk n, reeiver k finds ˆm k suh ha u n k ˆm k, s k, yk n T ɛ n for some s k REFERENCES ] T Q S Quek, M Peng, O Simone, and W Yu, Eds, Cloud Radio Aess Neworks: Priniples, Tehnoies, and Appliaions Cambridge Univ Press, Marh 07 ] Y Zhou and W Yu, Opimized bakhaul ompression for uplink loud radio aess nework, IEEE Journ Sel Are Comm, vol 3, no 6, pp , June 04 3] G Kramer, M Gaspar, and P Gupa, Cooperaive sraegies and apaiy heorems for relay neworks, IEEE Trans Inf Theory, vol 5, no 9, pp , Sep 005 4] A Sanderovih, O Somekh, H V Poor, and S Shamai, Uplink maro diversiy of limied bakhaul ellular nework, IEEE Trans Inf Theory, vol 55, no 8, pp , Aug 009 5] Y Zhou, Y Xu, W Yu, and J Chen, On he opimal fronhaul ompression and deoding sraegies for uplink loud radio aess neworks, IEEE Trans Inf Theory, vol 6, no, pp , De 06 6] A Sanderovih, S Shamai, Y Seinberg, and G Kramer, Communiaion via deenralized proessing, IEEE Trans Inf Theory, vol 54, no 7, pp , July 008 7] S H Lim, Y-H Kim, A E Gamal, and S-Y Chung, Noisy nework oding, IEEE Trans Inf Theory, vol 57, no 5, pp 33 35, May 0 8] S N Hong and G Caire, Compue-and-forward sraegies for ooperaive disribued anenna sysems, IEEE Trans Inf Theory, vol 59, no 9, pp , Sep 03 9] L Liu and W Yu, Join sparse beamforming and nework oding for downlink muli-hop loud radio aess neworks, Deember 06 0] L Liu, P Pail, and W Yu, An uplink-downlink dualiy for loud radio aess nework, in Pro IEEE In Symp Inf Theory, July 06, pp ] S Lim, K Kim, and Y-H Kim, Disribued deode-forward for relay neworks, arxiv: sit], Oober 05 ] A E Gamal and Y-H Kim, Nework Informaion Theory Cambridge, UK: Cambridge Univ Press, 0 3] A E Gamal, On informaion flow in relay neworks, in IEEE Na Teleom Conf, vol, November 98, pp D4 D44 067