An Information Theoretic Analysis for Adaptive-Network- Coded-Cooperation (ANCC) in Wireless Relay Networks

Transcription

1 ISIT 6, Seattle, US, July 9-4, 6 Iforatio Theoretic alysis for daptive-network- oded-ooperatio N i Wireless Relay Networks Xigkai ao ad Jig Li Tiffay Departet of lectrical ad oputer gieerig, Lehigh Uiversity, ethlehe, P 85 ail: {xib3, jigli}@ece.lehigh.edu bstract We coduct a iforatio theoretic aalysis for the adaptive-etwork-coded-cooperatio N protocol over large wireless relay etworks. The N protocol adaptively ecodes the data fro differet terials usig a sigle etwork code by atchig the istataeous etwork topoy with the code graph of a low-desity-parity-check LDP code. The ergodic capacity ad the outage probability of this protocol are aalyzed for both fiite ad ifiite etwork sizes, ad closedfor expressios are derived for the ifiite case. opariso with the existig protocols icludig repetitio ad space-tiecoded-cooperatio cofir that N is both low-coplexity ad high-perforace. I. INTRODUTION User cooperatio is a effective techique to cobat slow chael fadig i sigle-atea ulti-user wireless systes, where the trasit terials are allowed to share ateas to for a virtual atea array. arly researches focus o the protocols for the 3-terial basic sceario, such as repetitio, coded-cooperatio ad space-tie-coded-cooperatio ST. Motivated by the potetial to achieve a larger diversity gai, user cooperatio i the cotext of large wireless etworks is gaiig icreasig iterests 3. However, existig schees do ot scale well, ad are therefore iefficiet, expesive or wasteful to operate i a large etwork. or exaple, the repetitio protocol discussed i becoes itolerably badwidth iefficiet as the etwork size icreases. Spacetie-coded-cooperatio requires iter-user sychroizatio at the sybol level, which is techically challegig especially aog a large uber of distributed terials. With the recet advaces i cross-layer desigs, etwork codig 78, a extesio ad geeralizatio of routig, begis to fid it way to user cooperatio, a area predoiated by physical layer techoies. uber of papers eerged recetly, proposig etwork-codig-assisted cooperatio protocols 3-6. Most of the, however, use predefied, fixed etwork codes 3-5, with the ideal assuptio that the trasissio lik betwee ay two cooperatig users is free fro outage. Sice practical etworks cosistig of wireless fadig chaels are subject to rado lik failure ad topoy chage, fixed etwork codes will therefore break. or exaple, whe a relay fails to retrieve a packet eeded for its desigated codig operatio, the etire etwork code will have to abort. atural reedy to this critical issue is adaptatio. Havig a set of etwork codes, each apped to a possible etwork topoy, is a possibility, but works oly for sall etworks. ore practical ad efficiet solutio is to allow etwork codes This project is supported by NS uder Grat No , ad by the oowealth of Pesylvaia, Departet of ouity ad cooic Developet, through PIT. to be geerated distributedly, adaptively ad o-the-fly, as is proposed by the adaptive-etwork-coded-cooperatio N protocol i 6. The key idea here is to couple etworkso-graphs, i.e. istataeous etwork topoies described i graphs, with the well-kow class of codes-o-graphs, i.e. low-desity-parity-check LDP codes ad LDP-like codes 6. ig.. etwork graph describig etwork topoy. The correspodig bipartite code graph. thied code graph. To explai how N works, let us take a siple but geeralizable exaple where 5 users,,,, couicate to a coo destiatio D. or coveiece, we use packet ad sybol iterchageably. I the first phase, each user broadcasts its wireless data. ssue that for the tie beig, the iter-user coectivity is show i igure, where a directed edge idicates a successful trasissio destiatio D ot show. Without uch aipulatio, we ca trasfor this 5-ode etwork graph to a bipartite code graph that specifies a, 5 LDP-like code, as illustrated i igure. I the code graph, the 5 black circles, 5 boxes ad 5 white circles represet the source sybols origiated fro the 5 users, the etwork codig operatios to be perfored by these users, ad the relay sybols to be forwarded by these users, respectively. Hece, i the secod phase, each user ca copute ad forward the check-su of its correctlydecoded sybols, i a de-cetralized ad adaptive aer. sall bit-ap field will be icluded i each relay packet, so that the destiatio kows how the checks are fored ad ca correspodigly replicate the code graph ad perfor essage-passig decodig. To ake the essage-passig algorith effective, istead of perforig the check su o all the decoded sybols, each ode ca radoly pick oly afewsybols, thus thiig the code graph ad eliiatig the chaces for short cycles. or exaple, i igure, user ad user ay igore the source sybol fro user ad user whe perforig their respective codig operatio. The ew code graph, which ow happes to be free of legth-4 cycles, is show i igure. The efficiecy ad practicality of N is iitially deostrated i 6 by siulatios. This paper perfors iforatiotheoretic aalysis i ters of both capacity ad outage. or copariso purpose, we also aalyze repetitio-cooperatio /6/$. 6 I 79

2 ISIT 6, Seattle, US, July 9-4, 6 ad ST. We test the ipact of the uber of users, ad evaluate the scalability of these protocols with etwork size. I the boudary case where the etwork size goes to ifiity, closed-for capacity ad outage expressios are derived for all the three protocols. Our results provide a strog theoretic support for the excellet perforace of N. The reaider of the paper is orgaized as follows. Sectio II presets the syste odel. The capacity ad outage for N, ST ad repetitio are aalyzed i Sectio III, IV ad V, respectively. Nuerical evaluatio is discussed i Sectio VI. ially, Sectio VII cocludes the paper. II. SYSTM MODL We cosider terials trasittig data to a coo destiatio. The trasissio ca be divided ito two phases. I the first phase broadcastig phase, the terials broadcast their data i a roud Robi fashio. I the secod phase relayig phase, the terials help forward each other s data usig either repetitio-cooperatio, ST or N. We assue that all the trasissio chaels follow block fadig odel with chael fadig coefficiet α ad additive coplex Gaussia oise Z with zero ea ad variace N. We odel α as zero-ea, idepedet, circularly syetric coplex Gaussia rado variables with uit variace, so the agitudes α is Rayleigh distributed, ad the chael power u α is expoetially distributed with probability desity fuctio pdf p u y e y. y > The su of idepedet chael powers satisfies χ square distributio with pdf p χ y y! e y. y > The trasit SNR is defied as P/N, where P is the average power costrait of each terial. We assue that the fadig coefficiet α is kow to the receivers but ot the trasitters. This iplies that istataeous power adaptatio aog differet users or over differet tie slots is ot possible. Let decode-set, Di, deote the set of terials that ca successfully detect terial i s data, ad subscript i, d deote the chael fro terial i to the destiatio. III. DPTIV-NTWORK-ODD-OOPRTION N I the N protocol, i the secod phase, each terial radoly selects a sall uber of others data that have bee correctly decoded, adds the together i the biary doai, ad trasits the to the destiatio. t the destiatio, the data fro all the terials i both trasissio phases for a systeatic codeword, where the data trasitted i the broadcastig phase are take as the systeatic bits, ad the data trasitted i the relayig phase are cosidered as the parity bits. Thus, all the terials data are ecoded ito a sigle codeword of soe etwork code i a distributed aer. Ulike ST, i N, the sychroizatio aog differet terials is oly eeded at the packet level but ot the sybol level, which akes N practical i ipleetatio.. Mutual Iforatio I the N protocol, each packet occupies tie slots; hece, to oralize, both the utual iforatio ad trasit SNR eed to be scaled dow by a factor of. O the other had, each terial trasits a differet part of the etwork codeword, which aouts to utilizig idepedet chaels. The total utual iforatio obtaied i the secod phase equals the su of Shao forulas with terials istataeous SNR. urtherore, sice the etwork code provides equal protectio o average to all the terials, the utual iforatio carried across i the secod phase will cotributed evely to each terial. osequetly, we arrive at the followig result for the utual iforatio of N per terial I acc α i,d. rgodic apacity r α r,d. 3 ollowig the defiitio i 9, the ergodic capacity of a cooperatio protocol is give by the expectatio carried out with utual iforatio of rado chaels. Sice the utual iforatio of N is ot related to a certai decode-set see 3, we ca write the ergodic capacity of N directly as the expectatio of the utual iforatio o rado chaels as follows: acc I acc. 4 ro 3, ad sice the expectatio of a su of rado variables is equal to the su of the expectatios of these rado variables, 4 becoes acc y e y dy r y e y dy exp y e y dy, i / l, 5 where i. is the expoetial-itegral fuctio defied as ix e t x t dt. x > 6 It is worth otig that the capacity per terial of N is ot a fuctio of the etwork size. This is because the uderlyig assuptio here as well as i other papers i this area is that all the terials trasit over their desigated tie slots just as uch iforatio as proised by their istataeous sigle-chael capacities. Note that the etwork code i use is ot ecessarily optial, but each packet is assued to be optially chael coded. If the etwork code is also optial which will hold for large, the I acc P r α r,d. 7

3 ISIT 6, Seattle, US, July 9-4, 6. Outage Probability Outage probability, ΓR, is defied as the probability that a syste fails to istataeously support a predefied iforatio rate of R. ecause the utual iforatio of N is urelated to the decode-set, we ca write the outage probability fro the defiitio directly as Γ acc R PrI stcc <R, Pr α i,d α r,d <R. r 7 To evaluate 7, we defie f u r, 8 where u r is a expoetial rado variable with uit variace. ro the Jacobi law, the pdf of f ca be writte as p f y p u f y f y 4 l y e 4y /. 9 I additio, we defie f u r, whose pdf ca be writte as p f y p u f y f y, 4 l y e 4y /. Sice the pdf of a su of idepedet rado variables is equal to the covolutio of the idividual pdf, we have * deotes covolutio R {}}{ Γ acc R p f y p f y... p f y dy. This expressio is hard to siplify, so we resort to uerical itegral techiques to evaluate the outage for differet. The results are preseted i Sectio VI. Next, we cosider the outage probability whe teds to ifiity. ro the law of large ubers, we have li r α r,d α r,d, exp y exp y dy Therefore, we ca write i l. 3 li Γ accr Pr α i,d exp i <R, l exp R exp/i// l. 4 IV. SP-TIM-ODD-OOPRTION ST The ST protocol works like a geie-aided ulti-iput ulti-output MIMO syste, where i the relayig phase, all the terials that have received a clea copy of terial i s data utilize a suitable optial space-tie code to trasit this iforatio siultaeously.. Mutual Iforatio Siilar to N, each packet i ST takes up tie slots, so the utual iforatio eeds to be divided by. urtherore, assuig the space-tie codes i use are optial, the utual iforatio obtaied i the relayig phase eeds to be evaluated usig the Shao capacity forula with the eergy gathered fro all the terials i terial i th decode-set Di. To oralize the average trasit power, each terial i Di is allowed to use oly / Di of the eergy to trasit the space-tie code as copared to N, where deotes the cardiality of the set. The utual iforatio for each terial i the ST protocol is: I stcc α i,d α r,d. Di r Di 5. rgodic apacity Ulike the case of N, the utual iforatio of ST is related to the decode-set, so the ergodic capacity is give by the expectatio over the utual iforatio with rado chaels as follows: stcc I stcc I stcc Di Pr Di R, 6 where I stcc Di is the utual iforatio coditioed o the decode-set size, ad Pr Di R is the decode-set size probability give by Pr Di R Prr Di Prr / Di, e R e R. 7 Sice the chael powers i the first ter ad the secod ter of 5 follow the expoetial ad χ square distributios respectively, the expectatio of coditioal utual iforatio ca be derived as I stcc Di y exp ydy y y! e y dy. 8 quatig the trasissio rate to the ergodic capacity, we obtai the ergodic capacity of ST stcc as the solutio to 7

4 ISIT 6, Seattle, US, July 9-4, 6 the followig equatio with paraeter R: I stcc Di Pr Di R R, 9 which ca be coputed uerically. Specifically, whe teds to ifiity, use the law of large ubers, we have α r,d Di α r,d. r Di Thus, we ca write li stcc li I stcc, ye y dy, exp i / l.. Outage Probability ro the total probability law, the outage probability of ST Γ stcc R ca be coputed usig Γ stcc R Pr Di R Pr I stcc <R Di, where Pr Di R is give i 7. To calculate the coditioal outage probability Pr I stcc <R Di i, we take a siilar approach as i the case of N ad defie f 3 u r, 3 r whose pdf is give by p f3 y p χ f 3 y f3 y, l 4 y 4 y e 4y /!. 4 Sice the pdf of a su of idepedet rado variables equals the covolutio of their idepedet pdfs, we get Pr I stcc <R Di R p f y p f3 y dy, 8 l! R y6 y 4 y e 4y /r dy. 5 Whe the uber of terials teds to ifiity, gatherig 5, ad, we obtai li Γ accr, Pr α i,d exp <R, R /. 6 V. RPTITION-OOPRTION I repetitio-cooperatio, each user s data is expected to be repeated by all the other terials, ad tie slots are therefore assiged to each packet. Due to the possible iteruser outage, oly the terials i the decode-set Di are able to relay for i, so there are idle ad wasted tie slots. The utual iforatio i this case ca be writte as I rep α i,d α r,d. 7 r Di. rgodic apacity Siilar to the aalysis of ST, we equate the ergodic capacity to the trasissio rate ad get the capacity rep as the solutio to the followig equatio: I rep Di Pr Di rep rep, 8 where the decode-set size probability is Pr Di R e R e R. 9 Sice the fadig coefficiets for differet user chaels are i.i.d. distributed, the su of differet chael powers satisfies the χ-square distributio. The expectatio of the coditioal utual iforatio ca be writte as I rep Di y y! exp ydy. 3 Specifically, whe teds to ifiity, the utual iforatio of repetitio-cooperatio teds to zero. Thus, we ca write li rep. 3. Outage Probability or repetitio-cooperatio, sice Di is a rado set, fro the total probability law we get Γ rep R Pr Di Pr I rep <R Di, 3 where Pr Di is the decode-set size probability ad Pr I rep <R Di is the coditioal outage probability. ro 7, the coditioal outage probability satisfies Pr I rep <R Di R ad i the liit of ifiite, y exp y dy,! 33 li PrI rep <RPr <R. 34 7

5 ISIT 6, Seattle, US, July 9-4, 6 VI. NUMRIL RSULTS ollowig the aalytical expressios i the previous sectio, we provide i this sectio a uerical evaluatio of ad copariso betwee differet cooperatio protocols. The ergodic capacities are illustrated i igure. ro this figure, we observe that repetitio-cooperatio perfors the worst. Its ergodic capacity decreases with the icrease of uber of terials, ad evetually drops to zero as teds to ifiity. or ST, whe the uber of terials is sall, e.g., 5, it perfors very close to N. Sice its capacity icreases with whereas acc is ot a fuctio of, ST slightly outperfors N i large etworks at the cost of sigificatly icreased coplexity. We also observe that the capacity of ST for a oderate etwork size, say, already gets quite close to the asyptotic liit achieved whe. This suggests that the closedfor expressio i ca serve as a good predictio for the ergodic capacity of ST whe is large. or N, we see that it provides a costat capacity regardless of the uber of terials. Its perforace is cosistetly close to ST eve if ST has ifiite uber of terials. The outage probabilities are deostrated i igure 3 with trasissio rate threshold R /. It is ot surprisig that repetitio-cooperatio exhibits the worst outage perforace. The slope of the outage curves iproves with, idicatig a icreased diversity gai, but the curves are shifted further ad further to the right, idicatig a loss i codig gai which outweighs the diversity gai. oth ST ad N achieve uch better outage perforace tha repetitio or direct trasissio o cooperatio, ad their outage probabilities drop quickly with the icrease of SNR, showig a good diversity order. It is iterestig to ote that N slightly outperfors ST at sall, say 5, ad ST slightly outperfors N at large, say. ro the uerical results, it appears that is a threshold, where N ad ST exhibit about the sae outage probabilities. That the outage of ST iproves faster with tha N is i part attributed to the fact that the utual iforatio per terial of ST icreases with whereas the utual iforatio per terial of N is costat with respect to. VII. ONLUSIONS The siplicity, practicality ad efficiecy of adaptiveetwork-coded-cooperatio were deostrated by siulatios i 6. This paper provides a theoretical support for its excellet perforace by aalyzig its ergodic capacity ad outage probability. losed-for expressios are derived for the liitig case where the etwork size icreases without boud. or ay fiite etwork size where aalysis is ot tractable, we perfored uerical evaluatios. We have also copared N with repetitio ad space-tie-coded-cooperatio. We deostrate that N is superb to repetitio ad o par with ST i both capacity ad outage. osiderig that N obviates the eed for striget iter-user sychroizatio which is extreely costly i large wireless etworks, it is therefore a very attractive protocol i practice. Outage probability rgodic capacity bit/s/hz Rep 5 Rep Rep ST 5 ST ST N SNR d ig.. rgodic capacities for differet cooperatio protocols Direct Rep ST N SNR d ig. 3. Outage probabilities with rate threshold R /. RRNS M. Jaai,. Hedayat, T. Huter,. Nosratiia, oded cooperatio i wireless couicatios: space-tie trasissio ad iterative decodig, I Tras. Sigal Processig, vol. 5, NO., pp , eb 4, J. N. Laea, G. W. Worell, Distributed Space-Tie-oded protocols for exploitig cooperative diversity i wireless etworks, i I Trasactios o Iforatio Theory, vol. 49, NO., Oct 3, pp G. Kraer, M. Gastpar, ad P. Gupta, ooperative strategies ad capacity theores for relay etworks, to appear I Tras. Ifor. Theory, Hausl,. Schreckebach ad I. Oikooidis, Iterative etwork ad chael decodig o a taer graph, Proc. llerto of. o ou., otrol ad oputig Urbaa hapaig, IL, Sept Y. he, S. Kishore, ad J. Li, Wireless diversity through etwork codig, to appear, Proc.I Wireless ou. Networkig of., March, 6. 6 X. ao, ad J. Li, Matchig code-o-graph with etworks-o-graph: daptive etwork codig for wireless relay etworks, Proc. llerto of. o ou., otrol ad oputig Urbaa hapaig, IL, Sept R. hlswede, N. ai, S.-Y.R. Li ad R.W. Yeug, Network iforatio flow, I Tras. Ifor Theory, vol. 46, pp. 4-6,. 8 R. Koetter, M. Medard, algebraic approach to etwork codig, Tras. Networkig, Oct R. U. Nabar, H. ölcskei ad. W. Keubühler, adig relay chaels: Perforace liits ad space-tie sigal desig, I J. Selected reas i ou., Vol., NO. 6, ugust 4. 73