CLOUD Radio Access Network (C-RAN) is an emerging

Transcription

1 7402 EEE TRANSACTONS ON NFORMATON THEORY VOL. 62 NO. 2 DECEMBER 206 On the Optmal Fronthaul Compresson and Decodng Strateges for Uplnk Cloud Rado Access Networks Yuhan Zhou Member EEE YnfeXuStudent Member EEE WeYuFellow EEE and Jun Chen Senor Member EEE Abstract Ths paper nvestgates the compress-and-forward scheme for an uplnk cloud rado access network C-RAN model where mult-antenna base statons BSs are connected to a cloud-computng-based central processor CP va capacty-lmted fronthaul lnks. The BSs compress the receved sgnals wth Wyner-Zv codng and send the representaton bts to the CP; the CP performs the decodng of all the users messages. Under ths setup ths paper makes progress toward the optmal structure of the fronthaul compresson and CP decodng strateges for the compress-and-forward scheme n the C-RAN. On the CP decodng strategy desgn ths paper shows that under a sum fronthaul capacty constrant a generalzed successve decodng strategy of the quantzaton and user message codewords that allows arbtrary nterleaved order at the CP acheves the same rate regon as the optmal jont decodng. Furthermore t s shown that a practcal strategy of successvely decodng the quantzaton codewords frst then the user messages acheves the same maxmum sum rate as jont decodng under ndvdual fronthaul constrants. On the jont optmzaton of user transmsson and BS quantzaton strateges ths paper shows that f the nput dstrbutons are assumed to be Gaussan then under jont decodng the optmal quantzaton scheme for maxmzng the achevable rate regon s Gaussan. Moreover Gaussan nput and Gaussan quantzaton wth jont decodng acheve to wthn a constant gap of the capacty regon of the Gaussan multple-nput multple-output MMO uplnk C-RAN model. Fnally ths paper addresses the computatonal aspect of optmzng uplnk MMO C-RAN by showng that under fxed Gaussan nput the sum rate maxmzaton problem over the Gaussan quantzaton nose covarance matrces can be formulated as convex optmzaton problems thereby facltatng ts effcent soluton. Manuscrpt receved January 9 206; revsed August 9 206; accepted October Date of publcaton October 3 206; date of current verson November Ths work was supported by the Natural Scences and Engneerng Research Councl of Canada. Y. Xu was supported n part by a Grant from the Unversty Grants Commttee of the Hong Kong Specal Admnstratve Regon Chna under Project AoE/E- 02/08 and n part by the Natonal Natural Scence Foundaton of Chna under Grant Ths paper was presented n part at the 205 EEE nternatonal Symposum on nformaton Theory. Y. Zhou was wth The Edward S. Rogers Sr. Department of Electrcal and Computer Engneerng Unversty of Toronto Toronto ON M5S 3G4 Canada. He s now wth Qualcomm Technologes nc. San Dego CA 922 USA e-mal: yzhou@ece.utoronto.ca. Y. Xu was wth the School of nformaton Scence and Engneerng Southeast Unversty Nanjng Chna. He s now wth the Department of nformaton Engneerng nsttute of Network Codng The Chnese Unversty of Hong Kong Hong Kong e-mal: ynfexu@nc.cuhk.edu.hk. W. Yu s wth The Edward S. Rogers Sr. Department of Electrcal and Computer Engneerng Unversty of Toronto Toronto ON M5S 3G4 Canada e-mal: weyu@ece.utoronto.ca. J. Chen s wth the Department of Electrcal and Computer Engneerng McMaster Unversty Hamlton ON L8S 4L8 Canada e-mal: junchen@ece.mcmaster.ca. Communcated by O. Smeone Assocate Edtor for Communcatons. Dgtal Object dentfer 0.09/TT ndex Terms Cloud rado access network multple-access relay channel compress-and-forward fronthaul compresson jont decodng generalzed successve decodng.. NTRODUCTON CLOUD Rado Access Network C-RAN s an emergng moble network archtecture n whch base-statons BSs n multple cells are connected to a cloud-computng based central processor CP through wred/wreless fronthaul lnks. n the deployment of a C-RAN system the BSs degenerate nto remote antennas heads mplementng only rado functonaltes such as frequency up/down converson samplng flterng and power amplfcaton. The baseband operatons at the BSs are mgrated to the CP. The C-RAN model effectvely vrtualzes rado-access operatons such as the encodng and decodng of user nformaton and the optmzaton of rado resources []. Advanced jont multcell processng technques such as the coordnated mult-pont CoMP and network multple-nput multple-output MMO can be effcently supported by the C-RAN archtecture potentally enablng sgnfcantly hgher data rates than conventonal cellular networks [2]. Ths paper consders the uplnk of a MMO C-RAN system under fnte-capacty fronthaul constrants as shown n Fg. whch conssts of multple remote users sendng ndependent messages to the CP through multple BSs servng as relay nodes. Both the user termnals and the BSs are equpped wth multple antennas. The BSs and the CP are connected va noseless fronthaul lnks wth fnte capacty. Ths channel model can be thought of as a two-hop relay network wth an nterference channel between the users and the BSs followed by a noseless multple-access channel between the BSs and the CP. Ths paper assumes that a compress-andforward relayng strategy s employed n whch the relayng BSs perform dstrbuted lossy source codng to compress the receved sgnals and forward the representaton bts to the CP through dgtal fronthaul lnks and all the user messages are eventually decoded at the CP. The lossy source codng mplemented at BSs nvolves Wyner-Zv codng typcally consstng of quantzaton followed by bnnng n order to acheve hgh compresson effcency by leveragng the correlaton between the receved sgnals across dfferent BSs whch s dfferent from the pont-to-pont fronthaul compresson mplemented n today s conventonal C-RAN systems. A key queston n the desgn of compress-and-forward strategy n uplnk C-RAN s the optmal nput codng strategy EEE. Personal use s permtted but republcaton/redstrbuton requres EEE permsson. See for more nformaton.

2 ZHOU et al.: ON OPTMAL FRONTHAUL COMPRESSON AND DECODNG STRATEGES FOR UPLNK C-RANs 7403 Fg.. The uplnk C-RAN model under fnte-capacty fronthaul constrants. at the user termnals the optmal relayng strategy at the BSs and the optmal decodng strategy at the CP. Toward ths end ths paper restrcts attenton to the strategy of compressng the receved sgnals at the BSs then ether jont decodng of the quantzaton and message codewords smultaneously or generalzed successve decodng of the quantzaton and message codewords n some arbtrary order at the CP. Under ths assumpton ths paper makes the followng contrbutons toward revealng the structure of the optmal compress-andforward strategy. Frst motvated by the fact that successve decodng s much easer to mplement than jont decodng we seek to understand whether successve decodng at the CP can perform as well as jont decodng. Toward ths end ths paper shows that generalzed successve decodng ndeed acheves the same rate regon as jont decodng for an uplnk C-RAN model under a sum fronthaul constrant. Further although not necessarly so for the general rate regon f one focuses on maxmzng the sum rate the partcular strategy of successvely decodng the quantzaton codewords frst then the user messages acheves the optmal sum rate. Second we seek to understand the optmal nput dstrbuton and quantzaton schemes n uplnk C-RAN. Although t s well known that jont Gaussan strateges are not necessarly optmal ths paper shows that f we fx the nput dstrbuton to be Gaussan then the optmal quantzaton scheme s Gaussan under jont decodng and vce versa. Moreover jont Gaussan sgnalng can be shown to acheve the capacty regon of the Gaussan multple-nput multple-output MMO uplnk C-RAN model to wthn a constant gap. Fnally ths paper makes progress on the computatonal front by showng that under the jont Gaussan assumpton the optmzaton of the quantzaton covarance matrces for maxmzng the sum rate can be formulated as a convex optmzaton problem. These results suggest that jont Gaussan nput sgnalng and Gaussan quantzaton s a sutable strategy for the uplnk C-RAN. A. Related Work The achevable rate regon of compress-and-forward wth jont decodng for the uplnk C-RAN model was frst characterzed n [3] for a sngle-transmtter model then n [4] for the mult-transmtter case. However the number of rate constrants n the jont decodng rate regon grows exponentally wth the sze of the network [3 Proposton V.] whch makes the evaluaton of the achevable rate computatonally prohbtve. The achevable rate regon of the compressand-forward strategy wth practcal successve decodng n whch the quantzaton codewords are decoded frst then the user messages are decoded based on the recovered quantzaton codewords has also been studed for the uplnk C-RAN model [5 Theorem ]. One of the objectves of ths paper s to llustrate the relatonshp between jont decodng and successve decodng. n the exstng lterature the equvalence between these two decodng schemes s frst demonstrated for sngle-source sngle-destnaton and sngle-relay networks [6 Appendx 6C] then shown for sngle-source sngle-destnaton and multple-relay networks [7] under ether block-by-block forward decodng or block-by-block backward decodng. Ths paper further demonstrates that n the case of uplnk C-RAN whch s a multple-source sngle-destnaton multple-relay network the optmalty of successve decodng stll holds under sutable condtons. n general t s challengng to fnd the optmal jont nput and quantzaton nose dstrbutons that maxmze the achevable rate of the compress-and-forward scheme for uplnk C-RAN. Gaussan sgnalng s not necessarly optmal n partcular n a smple example of uplnk C-RAN wth one user and two BSs shown n [5] bnary nput s shown to outperform Gaussan nput for a broad range of sgnal-tonose ratos SNRs. However Gaussan nput and Gaussan quantzaton can be shown to be approxmately optmal. n fact the uplnk C-RAN model s an example of a general Gaussan relay network wth multple sources and a sngle destnaton for whch a generalzaton of compress-and-forward wth jont decodng referred to as nosy network codng scheme [8] [] and wth Gaussan nput and Gaussan quantzaton can be shown to acheve to wthn a constant gap to the nformaton theoretcal capacty of the overall network. nstead of usng nosy network codng our prevous work [2] shows that successve decodng can acheve the sum capacty of uplnk C-RAN to wthn constant gap f the fronthaul lnks are subjected to a sum capacty constrant. n ths work we further demonstrate that the compress-and-forward scheme wth jont decodng can acheve to wthn a constant gap to the entre capacty regon of the uplnk C-RAN model wth ndvdual fronthaul constrants; same s true for successve decodng under sutable condton. An mportant theoretcal result obtaned n ths paper s that f the nput dstrbutons of the uplnk C-RAN model are fxed to be Gaussan then Gaussan quantzer s n fact optmal under jont decodng. Fndng the optmal quantzaton for the C-RAN model s related to the mutual nformaton constrant problem [3] for whch entropy power nequalty s used to show that Gaussan quantzaton s optmal for a three-node relay network wth Gaussan nput. However t s challengng to extend ths approach to the uplnk C-RAN model whch has multple sources. Ths paper provdes a novel proof of the optmalty of Gaussan quantzaton based on the de Brujn dentty and the Fsher nformaton nequalty. The dea of the

3 7404 EEE TRANSACTONS ON NFORMATON THEORY VOL. 62 NO. 2 DECEMBER 206 proof s nspred by the connecton between the C-RAN model and the CEO problem n source codng [4] where a source s descrbed to a central unt by remote agents wth nosy observatons. The soluton to the CEO problem s known for the scalar Gaussan case [5] [6]; sgnfcant recent progress has been made n the vector case e.g. [7]. The smlarty between the uplnk C-RAN model and the CEO problem has been noted n [5] based on whch a capacty upper bound for the uplnk C-RAN model s establshed. n ths paper we use technques for establshng the outer bound for the Gaussan vector CEO problem [8] to prove the optmalty of Gaussan quantzaton. We also remark the connecton between ths quantzaton optmzaton problem and the nformaton bottleneck method [9] for whch jont Gaussan dstrbuton s shown to be Pareto optmal. The technque used n ths paper s a sgnfcantly smpler alternatve to the enhancement technque gven n [20]. Ths paper also makes progress n observng that the optmzaton of Gaussan quantzaton nose covarance matrces for maxmzng the weghted sum rate of uplnk C-RAN can be reformulated as a convex optmzaton problem. The quantzaton nose covarance optmzaton problem for uplnk C-RAN has been consdered extensvely n the lterature. Varous optmzaton algorthms have been developed to maxmze the achevable rates of the compress-and-forward scheme for the case of ether successve decodng of the quantzaton codewords followed by the user messages [2] [22] or jont decodng of the quantzaton codewords and user messages smultaneously [23]. n partcular a zero-dualty gap result has been shown for the weghted sum rate maxmzaton problem under a sum fronthaul capacty constrant n [2] based on a tme-sharng argument to facltate the algorthm desgn for searchng optmal quantzaton nose covarance matrces. However the optmzaton problems formulated n these works.e. [2] [23] are nherently nonconvex hence only locally convergent algorthms are obtaned. nstead ths paper provdes a convex formulaton of the problem that allows globally optmal Gaussan quantzaton nose covarance matrces to be found. Note that here the optmzaton of the quantzaton nose covarance matrx s performed under the fxed Gaussan nput. The jont optmzaton of the nput sgnal and quantzaton nose covarance matrces remans a computatonally challengng dffcult problem [24]. B. Man Contrbutons Ths paper establshes several nformaton theoretc results on the compress-and-forward scheme for the uplnk MMO C-RAN model wth fnte-capacty fronthaul lnks. A summary of our man contrbutons s as follows: Ths paper demonstrates that generalzed successve decodng for compress-and-forward whch allows the decodng of the quantzaton and user message codewords n an arbtrary order can acheve the same rate regon as jont decodng for compress-and-forward under a sum fronthaul capacty constrant. Further successve decodng of the quantzaton codewords frst then the user message codewords can acheve the same maxmum sum rate as jont decodng under ndvdual fronthaul constrants. Ths paper shows that under Gaussan nput and Gaussan quantzaton compress-and-forward wth jont decodng acheves to wthn a constant gap of the capacty regon of the uplnk MMO C-RAN model. Combnng wth the result above the same constant-gap result also holds for generalzed successve decodng under a sum fronthaul constrant and for successve decodng for sum rate maxmzaton. Ths paper shows that under fxed Gaussan nput Gaussan quantzaton maxmzes the achevable rate regon under jont decodng. Combnng wth the optmalty result for successve decodng ths also mples that under fxed Gaussan nput Gaussan quantzaton s optmal for generalzed successve decodng under a sum fronthaul constrant and for successve decodng for sum rate maxmzaton. Under jont Gaussan sgnalng and Gaussan quantzaton the optmzaton of quantzaton nose covarance matrces for maxmzng weghted sum rate under jont decodng and for maxmzng sum rate under successve decodng can be formulated as convex optmzaton problems whch facltate ther effcent soluton. C. Paper Organzaton and Notaton The rest of the paper s organzed as follows. Secton ntroduces the channel model for the uplnk MMO C-RAN and characterzes the achevable rate regons for compressand-forward schemes wth jont decodng and generalzed successve decodng respectvely. Secton demonstrates the rate-regon optmalty of generalzed successve decodng under a sum fronthaul constrant and the sum-rate optmalty of successve decodng. Secton V focuses on establshng the optmalty of Gaussan quantzers wth jont decodng under Gaussan nput. n addton Secton V also establshes the approxmate capacty of the uplnk MMO C-RAN model to wthn constant gap and shows the convex formulaton of the weghted sum rate maxmzaton problems over the quantzaton nose covarance matrces. Secton V concludes the paper. Notaton: Boldface letters denote vectors or matrces where context should make the dstncton clear. Superscrpts T and denote transpose operaton Hermtan transpose and matrx nverse operators; E[ ] and Tr denote expectaton and matrx trace operators; co denotes the convex closure operaton; p denotes the probablty dstrbuton functon n ths paper. We use X j = X X +...X j to denote a matrx wth j + columns for j. For a vector/matrx X X S denotes a vector/matrx wth elements whose ndces are elements of S. Gven matrces X...X L } dag X l } L denotes the block dagonal matrx formed wth X l on the dagonal. For random vectors X and Y JX Y denotes the Fsher nformaton matrx of X condtonal on Y; covx Y denotes the covarance matrx of X condtonal on Y.

4 ZHOU et al.: ON OPTMAL FRONTHAUL COMPRESSON AND DECODNG STRATEGES FOR UPLNK C-RANs ACHEVABLE RATE REGONS FOR UPLNK C-RAN A. Channel Model Ths paper consders an uplnk C-RAN model where K moble users communcate wth a CP through L BSs as shown n Fg.. The noseless dgtal fronthaul lnk connectng the BS l to the CP has the capacty of C l bts per complex dmenson. The fronthaul capacty C l s the maxmum long-term average throughput of the lth fronthaul lnk.e. lm n= n n C l C l wherec l represents the nstantaneous transmsson rate of the lth fronthaul lnk at the th tme slot. Each user termnal s equpped wth M antennas; each BS s equpped wth N antennas. Perfect channel state nformaton CS s assumed to be avalable to all the BSs and to the CP. For smple notaton we denote K = K } and L = L} n ths paper. Let X k C M be the sgnal transmtted by the kth user whch[ s subject ] to per-user transmt power constrant of P k.e. E X k X k P k. The sgnal receved at the lth BS can be expressed as K Y l = H lk X k + Z l l = 2...L k= where Z l CN0 l represents the addtve Gaussan nose for BS l and s ndependent across dfferent BSs and H lk denotes the complex channel matrx from user k to BS l. We consder the compress-and-forward scheme [25] [26] appled to the uplnk C-RAN system n whch the BSs compress the receved sgnals Y l and forward the quantzaton bts to the CP for decodng. At the CP the user messages are decoded usng ether jont decodng or some form of successve decodng. n jont decodng the quantzaton codewords and the message codewords are decoded smultaneously whereas n successve decodng the quantzaton codewords and messages are decoded successvely n some prescrbed order. Dfferent orderngs can potentally result n dfferent achevable rates. B. Achevable Rate-Fronthaul Regons for Jont Decodng Successve Decodng and Generalzed Successve Decodng n the followng we present the achevable rate-fronthaul regons of compress-and-forward wth jont decodng and dfferent forms of successve decodng. Proposton [3 Proposton V.]: For the uplnk C-RAN model shown n Fg. the achevable rate-fronthaul regon of compress-and-forward wth jont decodng PJD s the closure of the convex hull of all R R K C...C L R K +L R k < k T [ C l + satsfyng Y l ; Ŷ l X K ] + X T ; Ŷ S c X T c for all T K and S L for some product dstrbuton Kk= px k [ ] L pŷ l y l such that E X k X k P k for k =...K. 2 Note that for the uplnk C-RAN model the rate regon 2 gven by compress-and-forward wth jont decodng s dentcal to the rate regon of the nosy network codng scheme [9] whch s an extenson of the compress-and-forward scheme to the general multple access relay network by usng jont decodng at the recever and block Markov codng at the transmtters. As a more practcal decodng strategy successve decodng of quantzaton codewords frst and then the user messages at the CP can also be used n uplnk C-RAN. The followng proposton states the rate-fronthaul regon acheved by successve decodng. Proposton 2: [5 Theorem ]: For the uplnk C-RAN model shown n Fg. the achevable rate-fronthaul regon of compress-and-forward wth successve decodng PSD sthe closure of the convex hull of all R R K C...C L R+ K +L satsfyng R k < X T ; Ŷ L X T c T K 3 and k T Y S ; Ŷ S Ŷ S c < C l S L 4 for some product dstrbuton K k= p x k L [ ] pŷ l y l such that E X k X k P k for k =...K. Note that 3 s the multple-access rate regon 4 represents the Berger-Tung rate regon for dstrbuted lossy compresson [6 Theorem 2.] whle 2 ncorporates the jont decodng of the quantzaton codewords and the user messages. Because of ts lower decodng complexty successve decodng s usually preferred for practcal mplementaton of the uplnk C-RAN systems [2] [22]. Note that n the above strategy successve decodng apples only to the vector X k user message codewords and the vector Y l quantzaton codewords; the elements wthn vectors X k and Y l are stll decoded jontly. t s possble to mprove upon the successve decodng scheme by allowng arbtrary nterleaved decodng orders between quantzaton codewords and user message codewords. We call ths the generalzed successve decodng scheme n ths paper. The generalzed successve decodng scheme s frst suggested n [27] under the name of jont base-staton successve nterference cancelaton scheme. n such a successve decodng strategy the set of potental decodng orders ncludes all the permutatons of quantzaton and user message codewords. Denote π as a permutaton on the set of quantzaton and user message codewords Ŷ Ŷ 2...Ŷ L X X 2...X K. For a gven permutaton π the decodng order s gven by the ndex of the elements n π.e. π π2 πl + K. For example consder an uplnk C-RAN model as shown n Fg. wth 2 BSs and 2 users. f π = Ŷ X Ŷ 2 X 2 then the decodng of Ŷ 2 and X 2 can use both prevously decoded user messages and quantzaton codewords as sde nformaton. The resultng rate regon s

5 7406 EEE TRANSACTONS ON NFORMATON THEORY VOL. 62 NO. 2 DECEMBER 206 characterzed as R < X ; Ŷ R 2 < X 2 ; Ŷ Ŷ 2 X for some product dstrbuton px px 2 pŷ y pŷ 2 y 2 that satsfes C > Y ; Ŷ 6 C 2 > Y 2 ; Ŷ 2 Ŷ X. Let Xk Yl denote the ndces of user messages that are decoded before X k and Y l under the permutaton π respectvely. Lkewse let J Xk J Yl denote the ndces of quantzaton codewords that are decoded before X k and Y l under the permutaton π respectvely. The rate-fronthaul regon of generalzed successve decodng for uplnk C-RAN s stated n the followng proposton. Proposton 3: For the uplnk C-RAN model shown n Fg. the achevable rate-fronthaul regon of generalzed successve decodng wth decodng order π P GSD π sthe closure of the convex hull of all R R K C...C L R+ K +L satsfyng R k < X k ; Ŷ JXk X Xk k K 7 and C l > 5 Y l ; Ŷ l Ŷ JYl X Yl l L 8 for some product dstrbuton K k= p x k L [ ] pŷ l y l such that E X k X k P k for k =...K. The generalzed successve decodng regon PGSD s defned to be the closure of the convex hull of the unon of regons P GSD π over all possble permutaton π s.e. PGSD = co P GSD π. 9 π. OPTMALTY OF SUCCESSVE DECODNG n general we have P SD P GSD P JD. However successve decodng s more desrable than jont decodng not only because of ts lower complexty but also due to the fact that ts rate regon can be more easly evaluated. Thus there s a tradeoff between complexty and performance n desgnng decodng strateges for uplnk C-RAN. To further understand ths tradeoff ths secton establshes that: By allowng arbtrary decodng orders of quantzaton and message codewords the generalzed successve decodng actually acheves the same rate regon as jont decodng under a sum fronthaul constrant; 2 The practcal successve decodng strategy n whch the BSs decode the quantzaton codewords frst then the user messages actually acheves the same maxmum sum rate as jont decodng under ndvdual fronthaul constrants. A. Optmalty of Generalzed Successve Decodng Under a Sum Fronthaul Constrant Ths secton shows that n the specal case where the fronthaul lnks are subject to a sum capacty constrant generalzed successve decodng acheves the rate regon as jont decodng. n ths model the fronthaul capactes are constraned by L C l C and C l 0 justfable n stuatons where the fronthaul are mplemented n shared medum e.g. wreless fronthaul lnks as has been consdered n [2] and [2]. Under the sum fronthaul capacty constrant C the rate regons acheved by wth jont decodng R JDs s defned as R JDs R R K C...C L PJD = R...R K L C l C C l 0. 0 Lkewse the rate regon acheved wth generalzed successve decodng R GSDs s gven by R GSDs = R...R K R R K C...C L PGSD L C l C C l 0 }. The followng theorem states the man result of ths secton. Theorem : For the uplnk C-RAN model wth the sum fronthaul capacty constrant L C l C and C l 0 the rate regon acheved by generalzed successve decodng and jont codng are dentcal.e. R GSDs = R JDs. Proof: See Appendx A. The roadmap for the proof of Theorem shares the same dea as the characterzaton of the rate dstorton regon for the CEO problem under logarthmc loss [28] and the capacty regon for the multple-access channel [29] whch uses the propertes of submodular polyhedron see Appendx B. Specfcally n order to show R GSDs = R JDs we show that under fxed product dstrbuton K k= px k L pŷ l y l every extreme pont of the polyhedron R JDs C s domnated by the ponts n the polyhedron defned by R GSDs C. We conjecture that Theorem holds also for the case of ndvdual fronthaul capacty constrants. However n that case fndng the domnant faces of polyhedron PJD becomes much more dffcult t appears non-trval to extend the current proof to the case of ndvdual fronthaul constrants. B. Optmalty of Successve Decodng for Maxmzng Sum Rate As a specal nstance of generalzed successve decodng successve decodng reconstructs quantzaton codewords frst then user message codewords n a sequental order. n what follows we show that the optmal sum rate acheved by ths specal successve decodng s the same as that acheved by jont decodng.

6 ZHOU et al.: ON OPTMAL FRONTHAUL COMPRESSON AND DECODNG STRATEGES FOR UPLNK C-RANs 7407 Under fxed nput dstrbuton and fxed fronthaul capactes C l forl =...L the maxmum sum rate acheved by jont decodng R JDSU M s defned as K R JDSU M = max R k 2 k= s.t. R R K C...C L PJD. Lkewse the maxmum sum rate for successve decodng R SDSU M s gven by K R SDSU M = max R k 3 k= s.t. R R K C...C L PSD. The followng theorem demonstrates the optmalty of successve decodng for maxmzng uplnk C-RAN under ndvdual fronthaul constrants. Theorem 2: For the uplnk C-RAN model wth fronthaul capactes C l shown n Fg. the maxmum sum rates acheved by successve decodng and jont decodng are the same.e. R SDSU M = R JDSU M. Proof: See Appendx C. We remark that Theorem 2 can be thought as a generalzaton of a result n [7] that shows under block-byblock forward decodng the compress-and-forward scheme wth compresson-message successve decodng acheves the same maxmum rate as that wth compresson-message jont decodng for a sngle-source sngle-destnaton relay network. The uplnk C-RAN s a multple-source sngle-destnaton relay network. f all the user termnals are regarded as one super transmtter then t follows from [7] that successve decodng and jont decodng acheve the same maxmum sum rate. However the proof n [7] s qute complcated. n ths paper we provde an alternatve proof technque for showng the optmalty of successve decodng for sum rate maxmzaton n uplnk C-RAN. The new proof utlzes the propertes of submodular optmzaton whch s smpler than the proof provded n [7]. The proofs of Theorem 2 and Theorem llustrate the usefulness of submodular optmzaton n establshng ths type of results. t s remarked that successve decodng and jont decodng acheve the same sum rate but do not acheve the same rate regon. The achevable rate regon of generalzed successve decodng s n general larger than that of successve decodng. For example consder the compress-and-forward scheme for maxmzng the rate of user R only. The optmal decodng order should be X K\} Ŷ L X. Wth ths decodng order user can acheve larger rate than usng the decodng order of Ŷ L X K because the decoded user messages X 2 X 3...X K can serve as sde nformaton for the decodng of Ŷ L. n general to maxmze a weghted sum rate one needs to maxmze over L+K! orderngs for generalzed successve decodng. The man result of ths secton shows however that for maxmzng the sum rate n uplnk C-RAN successve decodng of the quantzaton codewords frst and then the user messages s optmal; ths reduces the search space consderably to L!K! decodng orders. V. UPLNK C-RAN WTH GAUSSAN NPUT AND GAUSSAN QUANTZATON n ths secton we specalze to the compress-and-forward scheme for uplnk C-RAN wth Gaussan nput sgnal at the users and Gaussan quantzaton at the BSs. Although t s known that jont Gaussan dstrbuton s suboptmal for uplnk C-RAN [5] Gaussan nput s desrable because t leads to achevable rate regons that can be easly evaluated. n the followng secton t s shown that wth Gaussan nput and Gaussan quantzaton compress-and-forward wth jont decodng can acheve the capacty regon of uplnk C-RAN to wthn a constant gap. The gap depends on the network sze but s ndependent of the channel gan matrx and the SNR. We further establsh the optmalty of Gaussan compresson at the relayng BSs for jont decodng f the nput s Gaussan. These results can be further extended to generalzed successve decodng under a sum fronthaul constrant and successve decodng for the maxmum sum rate. Addtonally under Gaussan sgnalng the optmzaton of quantzaton nose covarance matrces for weghted sum-rate maxmzaton under jont decodng and for sum rate maxmzaton under practcal successve decodng can be cast as convex optmzaton problems thereby facltatng ther effcent numercal soluton. Throughout ths secton we focus on the achevable rates under the fxed Gaussan nput and the fxed fronthaul capacty constrants C l for l =...L. A. Achevable Rate Regons Under Gaussan nput and Gaussan Quantzaton We let the nput dstrbuton be Gaussan.e. X k CN0 K k then evaluate the rate regons for the compressand-forward scheme wth jont decodng and successve decodng under Gaussan quantzaton denoted as R G JDGn and R G SDGn respectvely. Set L pŷ l y l CNy l Q l where Q l s the Gaussan quantzaton nose covarance matrx at the lth BS. Wth Gaussan nput and Gaussan quantzaton we have Y l ; Ŷ l X K = log l + Q l 4 Q l and X T ; Ŷ S c X T c H S c T K T H S = log c T + dag l + Q l } c. dag l + Q l } c 5 The achevable rate regon 2 for jont decodng can be evaluated as R k < [ C l log ] l + Q l Q l k T H S c T K T H S + log c T + dag l + Q l } c dag l + Q l } c 6 for all T K and S L.

7 7408 EEE TRANSACTONS ON NFORMATON THEORY VOL. 62 NO. 2 DECEMBER 206 Lkewse the achevable rate expresson 3 for successve decodng becomes H S c KK K H LK + dag l + Q l } R k < log dag l + Q l } k T 7 for all T K. n dervng the fronthaul constrant 4 we start wth evaluatng the mutual nformaton Y S ; Ŷ S Ŷ S c = X K Y S ; Ŷ S Ŷ S c X K ; Ŷ S Y S Ŷ S c = X K ; Ŷ S Ŷ S c + Y S ; Ŷ S X K Ŷ S c X K ; Ŷ S Y S Ŷ S c a = X K ; Ŷ S Ŷ S c + Y S ; Ŷ S X K b = X K ; Ŷ L X K ; Ŷ S c + Y l ; Ŷ l X K 8 for all S L where the equalty a follows from the fact that and Y S ; Ŷ S X K Ŷ S c = Y S ; Ŷ S X K 9 X K ; Ŷ S Y S Ŷ S c = 0 20 and equalty b follows from the fact that Y S ; Ŷ S X K = Y l ; Ŷ l X K. 2 The above equatons 9-2 follow from the Markov chan Ŷ Y X K Y j Ŷ j = j. We further evaluate the mutual nformaton expresson 8 wth Gaussan nput and Gaussan quantzaton whch yelds that Y S ; Ŷ S Ŷ S c H LK K K H LK + dag l + Q l } = log dag l + Q l } H S c KK K H S log c K + dag l + Q l } c dag l + Q l } c + log l + Q l Q l H LK K K H LK + dag l + Q l } = log H S c KK K H S c K + dag l + Q l } c log Q l C l. nstead of parameterzng the rate expressons over Q l as n above n ths secton we ntroduce the followng reparameterzaton whch s crucal for provng our man results. Defne B l = l + Q l. 22 We represent the rate regons of jont decodng and successve decodng n terms of B l n the followng. Proposton 4: For the uplnk C-RAN model shown n Fg. and under fxed Gaussan nput X K CN0 K K wth K K = dag K k } k K. The rate-fronthaul regon for jont decodng under Gaussan quantzaton PJDGn G s the closure of the convex hull of all R R K C...C L satsfyng R k < l C l log k T l B l + log c H lt B lh lt + KT 23 KT for all T [ K and ] S L forsome0 B l l where K T = E X T X T s the covarance matrx of X T andh lt denotes the channel matrx from X T to Y l. Furthermore under the fxed fronthaul capacty constrants C l for l =...L the rate regon acheved by jont decodng R G JDGn s defned as R G JDGn } = R...R K : R R K C...C L PJDGn G. 24 Proposton 5: For the uplnk C-RAN model shown n Fg. and under fxed Gaussan nput X K CN0 K K wth K K = dag K k } k K. The rate-fronthaul regon for successve decodng PSDGn G s the closure of the convex hull of all R R K C...C L satsfyng L H lt B lh lt + K T R k < log T K k T KT 25 and L H lk B lh lk + K K log + log H c lk B lh lk + KK l l B l < C l S L 26 [ ] for some 0 B l l wherek T = E X T X T s the covarance matrx of X T andh lt denotes the channel matrx from X T to Y l. Moreover under the fxed fronthaul capacty constrants C l for l =...L the rate regon acheved by

8 ZHOU et al.: ON OPTMAL FRONTHAUL COMPRESSON AND DECODNG STRATEGES FOR UPLNK C-RANs 7409 successve decodng R G SDGn s defned as R G SDGn = } R...R K : R R K C...C L PSDGn G. 27 B. Gaussan nput and Gaussan Quantzaton Acheve Capacty to wthn Constant Gap Wth Gaussan nput and Gaussan quantzaton the rate regon of jont decodng 23 can be shown to be wthn a constant gap to the capacty regon of uplnk C-RAN. Ths constant-gap result s stated n the followng theorem. Theorem 3: For any rate tuple R R 2...R K wthn the cut-set bound for uplnk C-RAN wth fxed fronthaul capactes of C l shown n Fg. the rate tuple R η R 2 η...r K η wth η = NL + M s achevable for compress-and-forward wth Gaussan nput Gaussan quantzaton and jont decodng where L s the number of BSs n the network M s the number of transmt antennas at user and N s the number of receve antennas at BS.e. R η R 2 η...r K η R G JDGn. Proof: See Appendx D. Although the uplnk C-RAN model s an example of a relay network for whch nosy network codng approach apples and t s known that compress-and-forward wth jont decodng acheves the same rate regon as nosy network codng for uplnk C-RAN we remark that Theorem 3 does not mmedately follow from the constant-gap optmalty result of nosy network codng [9]. The constant-gap optmalty of nosy network codng s proven for Gaussan relay networks whereas the uplnk C-RAN model contans fronthaul lnks whch are dgtal connectons and not Gaussan channels. Combnng wth our earler results on the optmalty of successve decodng constant-gap optmalty results can also be obtaned for compress-and-forward wth generalzed successve decodng and successve decodng. These results are summarzed n the followng corollary. Corollary : For the uplnk C-RAN model as shown n Fg. compress-and-forward wth generalzed successve decodng under Gaussan nput and Gaussan quantzaton acheves the capacty regon to wthn NL+ M bts per complex dmenson f the fronthaul lnks are subjected to a sum capacty constrant L C l C. Furthermore compressand-forward wth successve decodng under Gaussan nput and Gaussan quantzaton acheves the sum capacty of an uplnk C-RAN model wth ndvdual fronthaul constrants to wthn NL + MK bts per complex dmenson. C. Optmalty of Gaussan Quantzaton Under Jont Decodng For the Gaussan uplnk MMO C-RAN model t s known that Gaussan nput and Gaussan quantzaton are not jontly optmal [5]. However f the quantzaton nose s fxed as Gaussan then the optmal nput dstrbuton must be Gaussan. Ths s because the channel reduces to a conventonal Gaussan multple-access channel n ths case. The man result of ths secton s that the converse s also true.e. under fxed Gaussan nput Gaussan quantzaton actually maxmzes the achevable rate regon of the uplnk C-RAN model under jont decodng. Under fxed fronthaul capacty constrants C l for l =...L we let R JDGn denote the rate regon of jont decodng under Gaussan nput and optmal quantzaton. n the followng we frst defne Fsher nformaton and state the two man tools for provng ths result: the Brujn dentty and the Fsher nformaton nequalty. We then present the man theorem on the optmalty of Gaussan quantzaton for jont decodng.e. R G JDGn = R JDGn. Defnton : Let X Y be a par of random vectors wth jont probablty dstrbuton functon p x y. The Fsher nformaton matrx of X s defned as J X = E [ log p X log p X T]. 28 Lkewse the Fsher nformaton matrx of X condtonal on Y s defned as [ J X Y = E log p X Y log p X Y T]. 29 Lemma : Fsher nformaton nequalty [30] [8 Lemma 2]: Let U X be an arbtrary complex random vector where the condtonal Fsher nformaton of X condtoned on U exsts. We have log πej X U h X U. 30 Lemma 2 Brujn dentty [3] [8 Lemma 3]: Let V V 2 be an arbtrary random vector wth fnte second moments and N be a zero-mean Gaussan random vector wth covarance N. Assume V V 2 and N are ndependent. We have cov V 2 V V 2 + N = N N J V 2 + N V N. 3 Theorem 4: For the uplnk C-RAN under fxed Gaussan nput dstrbuton and assumng jont decodng Gaussan quantzaton s optmal.e. R G JDGn = R JDGn. Proof: Recall that the achevable rate regon of the compress-and-forward scheme under jont decodng s gven by the set of R...R K derved from 2 under the jont dstrbuton p x...x K y...y L ŷ...ŷ L K L L = p x k p y l x...x K p ŷ l y l. 32 k= For fxed Gaussan nput X K CN0 K K and fxed L pŷ l y l choose B l wth 0 B l l such that cov Y l X K Ŷ l = l l B l l l = L. We proceed to show that the achevable rate regon as gven by 23 wth a Gaussan L pŷ l y l CNY l Q l where Q l = B l l s as large as that of 2 under Gaussan nput.

9 740 EEE TRANSACTONS ON NFORMATON THEORY VOL. 62 NO. 2 DECEMBER 206 Frst note that Y l ; Ŷ l X K = log πe l h Y l X K Ŷ l Y l X K Ŷ l log πe l log πe cov l = log l = L 33 l B l where we use the fact that Gaussan dstrbuton maxmzes dfferental entropy. Moreover we have X T ; Ŷ S c X T c = h X T h X T X T c Ŷ S c log K T log J X T X T c Ŷ S c where the nequalty s due to Lemma. Snce Y S c = H S c T X T + H S c T cx T c + Z S c t follows from the MMSE estmaton of Gaussan random vectors that X T where G T l = KT = E [X T X T c Y S c] + N T S c = c G T l Yl H lt cx T c + NT S c + j S c H jt j H jt H lt l and N T S c CN 0 N wth covarance matrx N = KT + H lt l H lt. 34 c Here E [X T X T c Y S c] s the MMSE estmator of X T from X T c Y S c. The error n estmaton s N T S c and the MMSE matrx s N. By the matrx complementary dentty between Fsher nformaton matrx and MMSE n Lemma 2 we have J X T X T c Ŷ S c = N N cov G T l Y l H lt cx T c X K Ŷ S c N c = N = N = N N cov N c H lt G T l Y l X K Ŷ S c [ c G T l cov c l B l H lt = K T + c H lt B lh lt. N Y l X K Ŷ l G T l ] N Therefore X T ; Ŷ S c X T c log = log JX T X T c Ŷ S c KT KT + c H lt B lh lt KT 35 for all T K and S L. Combnng 33 and 35 we conclude that R G JDGn as derved from 23 s as large as R JDGn. Therefore RG JDGn = R JDGn. D. Optmzaton of Gaussan nput and Gaussan Quantzaton Nose Covarance Matrces Ths secton addresses the numercal optmzaton of the Gaussan nput and quantzaton nose covarance matrces for uplnk MMO C-RAN under gven fronthaul capacty constrants. Frst we note that even when restrctng to Gaussan nput and Gaussan quantzaton the jont optmzaton of nput and quantzaton nose covarance matrces s stll a challengng problem for the uplnk MMO C-RAN. However f we fx the quantzaton nose covarance then the nput optmzaton reduces to that of optmzng a conventonal Gaussan multple-access channel. n partcular the problem of maxmzng the weghted sum rate can be formulated as a convex optmzaton whch can be readly solved [32]. Conversely f we fx the transmt covarance matrx the optmzaton of quantzaton nose covarance can n some cases be formulated as convex optmzaton. The key enablng fact s the reparameterzaton n term of B l 22 nstead of drect optmzaton over Q l. Consder frst the case of jont decodng. Usng 23 under the fxed C l for l =...L the weghted sum rate maxmzaton problem can be formulated over R k B l } as follows: max R k B l s.t. K μ k R k k= R k log c H lt B lh lt + KT k T KT + l C l log T K S L l B l 0 B l l l L 36 where μ k represents the weght assocated wth user k whch s typcally determned from upper layer protocols. The key observaton s that the above problem s convex n R k B l }. However we also note that because of jont decodng the number of constrants s exponental n the sze of the network. Consequently the above optmzaton problem can only be solved for small networks n practce. Note that the above formulaton consders the optmzaton of nstantaneous achevable rates R k under nstantaneous fronthaul capacty constrants C l n a fxed tme slot. The soluton obtaned however also apples to the more general case

10 ZHOU et al.: ON OPTMAL FRONTHAUL COMPRESSON AND DECODNG STRATEGES FOR UPLNK C-RANs 74 of optmzng the weghted sum rates under weghted sum fronthaul constrant e.g. L ν l C l C. Ths s because f we consder a slghtly more general formulaton of optmzng an objectve of max R k B l C l k= K μ k R k γ L ν l C l 37 under the same constrants as n 36 and L ν l C l C. Such an optmzaton problem s convex so tme-sharng s not needed. For ths reason the rest of ths secton consders the formulaton wth nstantaneous rates only. We now consder the weghted sum-rate maxmzaton problem for the case of successve decodng of the quantzaton codewords followed by the user messages. However the drect characterzaton of successve decodng rate does not gve rse to a convex formulaton. Nevertheless for the specal case of maxmzng the sum rate.e. wth μ = =μ K = usng Theorem 2 whch shows that successve decodng acheves the same maxmum sum rate as jont decodng the sum-rate maxmzaton problem wth successve decodng can be equvalently formulated as follows: Theorem 5: For the uplnk C-RAN model wth ndvdual fronthaul capacty constrant C l asshownnfg.thesum rate maxmzaton problem under successve decodng can be formulated as the followng convex problem: max RB l R s.t. R l C l log l B l + log c H lt B lh lt + KK S L KK 0 B l l l L. 38 Further f the fronthaul lnks are subject to a sum capacty constrant of C the sum rate maxmzaton problem can be formulated as the followng convex problem: max RB l R L H lk B lh lk + K K s.t. R log KK L l R + log C l B l 0 B l l l L. 39 We remark that the formulaton for uplnk C-RAN wth ndvdual fronthaul capactes 38 has exponental number of constrants because the CP n effect needs to search over L! dfferent decodng orders of quantzaton codewords at the BSs. n practcal mplementaton a heurstc method can be used to determne the decodng orders of quantzaton codewords for avodng the exponental search [24] [33]. Alternatvely f the C-RAN has a sum fronthaul constrant then the number of constrants s lnear n network sze because we only need to consder the case of S = L and S = n 38. Consequently the resultng quantzaton nose covarance optmzaton problem 39 can be solved n polynomal tme. Note that convexty s a key advantage of the above problem formulatons as compared to prevous approaches n the lterature e.g. [2] [22] that parameterze the optmzaton problem over the quantzaton nose covarance Q l whch leads to a nonconvex formulaton. We emphasze the mportance of Gaussan nput for the convex formulaton n Theorem 5. Suppose that both nput sgnal X K and compressed sgnal Ŷ l are dscrete random vectors wth fnte alphabet. For fxed nput dstrbuton the sum-rate maxmzaton problem under the sum fronthaul constrant can be wrtten as max X K ; Ŷ L pŷ l y l s.t. Y L ; Ŷ L C p ŷ l y l 0 ŷ l p ŷl y l = l L. 40 The above problem can be thought as a varant of the nformaton bottleneck method [9] whch can be solved by a generalzed Blahut-Armoto BA algorthm [34] [35]. However due to the non-convex nature of problem 40 the generalzed BA algorthm can only converge to a local optmum. V. CONCLUSON Ths paper provdes a number of nformaton theoretcal results on the optmal compress-and-forward scheme for the uplnk MMO C-RAN model where the BSs are connected to a CP through noseless fronthaul lnks of lmted capactes. t s shown that the generalzed successve decodng scheme whch allows arbtrary decodng orders between quantzaton and user message codewords can acheve the same rate regon as jont decodng under a sum fronthaul constrant. Moreover the practcal successve decodng of the quantzaton codewords followed by the user messages s shown to acheve the same maxmum sum rate as jont decodng under ndvdual fronthaul constrants. n addton f the nput dstrbuton s assumed to be Gaussan t s shown that Gaussan quantzaton maxmzes the achevable rate regon of jont decodng. Wth Gaussan nput sgnalng the optmzaton of Gaussan quantzaton for maxmzng the weghted sum rate under jont decodng and the sum rate under successve decodng can be cast as convex optmzaton problems whch facltates effcent numercal soluton. Fnally Gaussan nput and Gaussan quantzaton acheve the capacty regon of the uplnk C-RAN model to wthn constant gap. Collectvely these results provde justfcatons for the practcal choce of usng Gaussan-lke nput sgnals at the user termnals Gaussan-lke quantzaton at the relayng BSs and successve decodng of quantzaton codewords followed by user messages at the CP for mplementng uplnk MMO C-RAN.

11 742 EEE TRANSACTONS ON NFORMATON THEORY VOL. 62 NO. 2 DECEMBER 206 V. APPENDX A OPTMALTY OF GENERALZED SUCCESSVE DECODNG n ths appendx we prove Theorem whch states the equvalence between generalzed successve decodng and jont decodng under a sum-capacty fronthaul constrant. We begn by ntroducng an outer bound for the achevable rate regon of jont decodng under a sum fronthaul constrant. Under the sum fronthaul capacty constrant defne the rate-fronthaul regon for jont decodng PJDs o as the closure of the convex hull of all R R 2...R K C satsfyng R k < mn C Y l ; Ŷ l X K k T X T ; Ŷ L X T c } T K 4 C > Y l ; Ŷ l X K for some product dstrbuton K k= p x k L pŷ l y l. Under fxed sum fronthaul constrant C defne the regon R o JDs as follows } R o JDs R =...R K : R R K C PJDs o. 42 Note that the rate regon R o JDs s an outer bound for jont decodng rate regon 0 because only the constrants correspondng to S = and S = L are ncluded. These constrants turn out to be the only actve ones under the sum fronthaul constrant L C l C and C l 0. Under the sum fronthaul constrant the generalzed successve decodng regon P GSDs π for decodng order π can be derved from 2 by lettng L C l = C. More specfcally P GSDs π s the closure of the convex hull of all R R 2...R K C satsfyng R k < X k ; Ŷ JXk X Xk k K L 43 C > Y l ; Ŷ l Ŷ JYl X Yl for some product dstrbuton K k= p x k L pŷ l y l where Xk Yl are the ndces of user messages that are decoded before X k and Y l under the permutaton π andj Xk J Yl are the ndces of the quantzaton codewords that are decoded before X k and Y l under decodng order π. Defne PGSDs to be the closure of the convex hull of all P GSDsπ s over decodng order π s.e. PGSDs = co P GSDs π. π We say a pont R...R K C s domnated by a pont n PGSDS f there exsts some R...R K C n PGSDs for whch R k R k for k = 2...K andc C. Gven the defntons of R GSDs R JDs and Ro JDs ts easy to see that R GSDs R JDs R o JDs. To show R GSDs = R JDs t suffces to show Ro JDs R GSDs whch s equvalent to showng that f a pont R R 2...R K C PJDs o then the same pont R R 2...R K C PGSDs also. To show ths t suffces to show that for any fxed product dstrbuton K k= p x k L pŷ l y l and fxed C each extreme pont R...R K C as defned by 4 s domnated by a pont n PGSDs wth the average sum fronthaul capacty requrement at most C. To ths end defne a set functon f : 2 K R as follows: f T := mn C Y l ; Ŷ l X K X T ; Ŷ L X T c } for each T K. t can be verfed that the functon f s a submodular functon Appendx B Lemma 3. By constructon R R 2...R K as defned by 42 satsfes R k f T k T whch s a submodular polyhedron assocated wth f. t follows by basc results n submodular optmzaton Appendx V Proposton 6 that for a lnear orderng 2 K on the set K an extreme pont of R JDs can be computed as follows R j = f... j } f... j }. Furthermore the extreme ponts of R o JDs can be enumerated over all the orderngs of K. Each orderng of K s analyzed n the same manner hence for notatonal smplcty we only consder the natural orderng j = j n the followng proof. By constructon R j = mn C } Y l ; Ŷ l X K X j ; Ŷ L X K j+ mn C } Y l ; Ŷ l X K X j ; Ŷ L X K j. 44 Due to the fact that X j ; Ŷ L X K j+ X j ; Ŷ L X K j for some product dstrbuton K k= p x k L pŷ l y l equaton 44 can yeld two dfferent results. Case : the frst term C Y l; Ŷ l X K n the mnma n equaton 44 s not actve for any j; Case2:theterm C Y l; Ŷ l X K s actve startng wth some ndex j. Case holds f C X K ; Ŷ L + Y l ; Ŷ l X K. n ths case the resultng extreme pont r JD = R R 2... R K C satsfes R j = X j ; Ŷ L X K j+ for j = 2...K R K = X K ; Ŷ L C = X K ; Ŷ L + Y l ; Ŷ l X K. Consder successve decodng wth the decodng order Ŷ L X K X. The extreme pont R...R K C PGSDs correspondng to ths decodng order s R j = X j ; Ŷ L X K j+ for j = 2...K R K = X K ; Ŷ L C = Y L ; Ŷ L.

12 ZHOU et al.: ON OPTMAL FRONTHAUL COMPRESSON AND DECODNG STRATEGES FOR UPLNK C-RANs 743 Followng the Markov chan Ŷ Y X K Y j Ŷ j = j t can be shown that Y l ; Ŷ l X K + X K ; Ŷ L = Y L ; Ŷ L. Clearly r JD can be acheved by the decodng order of Ŷ L X K X. Thus r JD s domnated by a pont n PGSDs. Case 2 holds f C X K ; Ŷ L + Y l ; Ŷ l X K. We let X j = for < j and assume that X j ; Ŷ L X K j C Y l ; Ŷ l X K and C Y l ; Ŷ l X K X j ; Ŷ L X K j+ for some j K. The resultng extreme pont r 2 JD = R R 2... R K C satsfes R = X ; Ŷ L X+ K for < j [ R = C ] + Y l ; Ŷ l X K X j ; Ŷ L X K for = j R = 0 for > j C = X j ; Ŷ L X K j+ + Y l ; Ŷ l X K where [ ] + means max 0}. Note that users wth ndex > j are nactve and are essentally removed from the network. n ths case the rate-fronthaul tuple does not correspond to a specfc corner pont obtaned wth a specfc generalzed successve decodng order but that t les on the convex-hull of two corner ponts of two dfferent generalzed successve decodng orders. To obtan a vsualzaton on Case 2 the rate-fronthaul regon for a two-user C-RAN model under a fxed jont dstrbuton p x x 2 y y 2 ŷ ŷ 2 s llustrated n Fg. 2. n the case of K = j = 2 t s shown that the ratefronthaul tuple r 2 JD les on the convex-hull of two corner ponts r and r 2. To prove the statement mathematcally we consder generalzed successve decodng wth the followng two dfferent decodng orders: Decodng order satsfes X K... X j+ Ŷ L X j... X. The extreme pont r GSD = R...R K C of PGSDs correspondng to Decodng order satsfes R = X ; Ŷ L X+ K for j R = 0 for > j C = Y L ; Ŷ L X K j+ Fg. 2. An llustraton of the rate-fronthaul tuple n Case 2 n Appendx A wth a two-user C-RAN model under a fxed jont dstrbuton p x x 2 y y 2 ŷ ŷ 2. where C represents the requred fronthaul capacty n order to acheve the above rate tuple R...R K wth decodng order. Decodng order 2 s X K... X j Ŷ L X j... X. The extreme pont r 2 GSD = R 2...R2 K C2 of PGSDs correspondng to Decodng order 2 satsfes R 2 = X ; Ŷ L X+ K for < j R 2 = 0 for j C 2 = Y L ; Ŷ L X K j where C 2 represents the requred fronthaul capacty n order to acheve the above rate tuple R 2...R2 K wth decodng order 2. Observe that the rate tuples R...R K and R2...R2 K gven by above two decodng orders dfferent at only the jth component where R j = X j ; Ŷ L X K j+ and R 2 j = 0 and R = R 2 = R for all < j. Now choose a parameter θ such that C Y l ; Ŷ l X K X j ; Ŷ L X K j θ =. X j ; Ŷ L X K j+ 45