Omnidirectional Relay in Wireless Networks

1 Omnidirectional Relay in W ireless Netw orks Liang-Liang Xie Departmen t of E lectrical an d Com puter En gineerin g University of W aterloo, W aterlo o, O N, Canada N2 L 3G1 Email: llxie@ece. uwaterloo.ca Abstract For wireless networks with multiple so urces, an o mnidire ctional relay scheme is dev eloped, wher e each node can simultan eously relay d ifferent messages in different directions. This is accomp lished by the decod e-and- forward relay strategy , with each relay binning the multiple m essages to be tran smitted, in the same spirit of network coding. Specially for the all-sou rce all-cast pr oblem, where each n ode is an independe nt source to be transmitted to all th e o ther n odes, this schem e co mpletely eliminates interferen ce in the whole network, an d the signa l transmitted by any nod e can be used by any other node. For networks with some kind of symmetr y , assuming no beamf orming is to be perfo rmed, this omnidire ctional relay scheme is cap able of achieving the maximu m achiev able rate. I . I N T RO D U C T I O N In wireless networking, relay is a way of expanding communication range or in creasing communication rate, with the help of other nodes. As s uch, more nodes are in v olved and more signals will be transmi tted. It is therefore i mportant to d esign and coordi nate these si gnals to maximize the cooperation and mini mize the in terference. Between t he two fundam ental relay strategies prop osed in [1], especially , the decode-and-forward strategy enables t he destinati on node t o fully enjoy the transmi tted power of bot h t he so urce node and th e relay nod e. Thi s is still realizable when multiple r elays are introduced to help the destination [2], [3], [ 4], and interference can be com pletely eliminated for arbitrarily lar ge networks. Howe ver , the situation is much more compl icated when t here are multiple sou rces in the network [5]. Unl ike t he case of a single source wh ere all nodes are essentially transmitti ng the same i nformation, mult iple sources seem ine vitably resu lt in interference. N e vertheless, studies of the two-way relay channel [6], [7] hav e indicated t he poss ibility of no i nterference even i f there are more than one sources. In this paper , we develop an omnid irectional relay scheme for wireless networks with multiple sources, where, each node can sim ultaneously relay d iff erent messages in di f ferent directions . 2 This is accomplis hed by binn ing mu ltiple messages at each relay , as a generalization of the scheme prop osed in [7], in the same spirit of n etwork coding [8]. The basi c idea of network codi ng [8] can be explained with the fol lowing example. Suppose that nod e A wants to send ou t two b its of information b 1 and b 2 , with b 1 to nod e B , and b 2 to node C . Howe ver , if node B already knows b 2 and no de C already knows b 1 , then th is can be accomplished by j ust sendin g out one bit b 1 ⊕ b 2 to both node B and no de C , since node B can recover b 1 by comput ing b 2 ⊕ ( b 1 ⊕ b 2 ) = b 1 , and node C can recov er b 2 by comput ing b 1 ⊕ ( b 1 ⊕ b 2 ) = b 2 . This scheme can be generalized with the technique of binning [7]. Consider the problem that node A wants t o send out two messages w 1 and w 2 , with w 1 to nod e B , and w 2 to node C , where, w 1 can take M 1 diffe rent values and w 2 can take M 2 diffe rent values, and possibly , M 1 6 = M 2 . Sim ilarly , assum e t hat n ode B already knows t he t rue value of w 2 , and node C already knows th e true value of w 1 . Inst ead of sending out both t he m essages ( w 1 , w 2 ) , wh ich can be any of the M 1 M 2 diffe rent vectors, n ode A can throw these vectors into M bin s, wi th M = max { M 1 , M 2 } , and send o ut t he index of the bin that contai ns th e true ve ctor . In this wa y , node A only needs to send out a message wit h M different values. It can be easily checked that when M ≥ max { M 1 , M 2 } , i t i s p ossible to b in the M 1 M 2 diffe rent vec tors of ( w 1 , w 2 ) in s uch a way that in each bin, no t wo vectors contain the same w 1 or the same w 2 . Therefore, knowing the true value of w 2 , and the bin that contains the true value of ( w 1 , w 2 ) , no de B can uniq uely determine the true value of w 1 . Sim ilarly , node C can uni quely determin e the true value of w 2 . The above binning s cheme can be easily generalized to send any num ber of messages. In the context of wireless relay networks, node A can be a relay t hat wants to forward d iffe rent messages to differe nt nod es. W ith the binning technique, n ode A only needs to send one sign al representing the bi n index, from which, dif ferent receivers can pick up dif ferent m essages based on their d iffe rent a pr iori k nowledge of the m essages. Furthermore, wit h this bin ning scheme, it is also shown in [7 ] that ev ery recei ver can fully exploit all the signal power , as if node A is only sendi ng those messages unknown to it . Node A can also use other ways to relay mu ltiple messages, e.g., by su perposition coding. It can first encode each message individually by a signal, and then superpose them together into a l ayered signal to transmit . Upon recei ving this layered signal, each recei ver can pick out the layers that correspond to the unknown messages, by deleti ng the layers that correspond to the messages already known. Compared to the binni ng scheme, an obvious drawback o f this superposition scheme is that the t otal t ransmit power of node A has to be clearly di vided among the messages, and each receive r can only exploit the part that is used for its unknown messages. 3 Howe ver , thi s way of clearly layering diff erent messages m akes i t easier to establish cooperation between different transmit ters. For example, to send the same message to a commo n receiv er , beamforming or coherent transm ission can be established between two transm itters so that the recei ved power can be boosted. On the other hand, this is no t so easy to realize with the bi nning scheme un less the two transmitters are sending exactly the same set of messages. Using superposition coding to establish coherent transmissi on was originally proposed in [1] for the relay channel. It was later extended to the case with mul tiple relays [2], [3], [4], and to t he two-way relay channel [6]. It can also be applied to a general framew ork wi th multipl e sources, relays and destinations [5]. Howev er , the correspondi ng achiev able rate regions become extremely messy for general networks, when there are too man y l ayers of signals to consi der . In this paper , we only consi der the binning technique in the omnidirectional relay s cheme. As a special applicati on which may be the best t o demonst rate t he benefit of this binning scheme, we consider the all-source all-cast problem, where each node is an independent s ource, to be sent to all the o ther nodes. W e will s how that for such probl ems, it is p ossible to completely eliminate interference in the network, and each node will enjoy the po wer transmitted b y all the other no des. The remainder of the paper is organized as the following. In Section II, we introdu ce a general framework of om nidirectional relay with arbitrary source-destinati on distributions in mind. Starting from Section III, we will focus on th e all-source all-cast problem. First, a special version of the omni directional relay scheme is developed in Section III for the all-source all - cast problem. Then a key technical lemma is presented i n Section IV, before we prove some achie vability results in Section V. Finally , some concluding remarks are presented in Section VI. I I . A N O M N I D I R E C T I O NA L R E L A Y S C H E M E Consider a wireless network of n nod es N = { 1 , 2 , . . . , n } . Consider the following A WGN wireless network channel model: Y j ( t ) = X i ∈N i 6 = j g i,j X i ( t ) + Z j ( t ) , ∀ j ∈ N , t = 1 , 2 , . . . (1) where, X i ( t ) ∈ C 1 and Y i ( t ) ∈ C 1 respectiv ely denote the signals s ent and received by Node i ∈ N at time t ; { g i,j ∈ C 1 : i 6 = j } denote th e sig nal attenuation gains; and Z i ( t ) is zero-mean complex Gaus sian noise wi th va riance N . Note t hat we are consi dering a ful l-duplex model , i.e., nodes can transmit and receive signals at the same tim e. Howe ver , it wil l be clear that the main results of this paper can be easily extended to half-duplex models. 4 Consider the networking problem wh ere each node i ∈ N wants t o send the sam e information at rate R i (can b e zero) to all the nodes in a subset T i ⊂ N . Or reversely , each node i ∈ N wants to recei ve the information sent b y all the nodes i n so me s ubset S i ⊂ N . T o achie ve this, we desig n an omnidirectional relay scheme as t he foll owing. W e choo se a sequence of decode-sets and encode-sets for each n ode in N in t he following order . First, for each node i ∈ N , choose a subset of N \{ i } as its 1-hop decode-set D i (1) , and then cho ose a subset of D i (1) as it s 1-hop encode-set E i (1) . That is, E i (1) ⊆ D i (1) ⊆ N \{ i } . Then, for each node i ∈ N , choose i ts 2-hop decode-set and encode-set as D i (2) ⊆ N \  { i } ∪ D i (1)  E i (2) ⊆  D i (1) ∪ D i (2)  \E i (1) Sequentially , for k = 3 , 4 , . . . , L , where L is some selected finite integer , node i ’ s k -hop decode- set and encode-set are chosen as D i ( k ) ⊆ N \  { i } ∪ D i (1) ∪ · · · ∪ D i ( k − 1)  E i ( k ) ⊆  D i (1) ∪ · · · ∪ D i ( k )  \  E i (1) ∪ · · · ∪ E i ( k − 1)  W e use block Markov coding. Consider B blocks of equal l ength, and in each block b = 1 , 2 , . . . , B , denote the message of node i by w i ( b ) , whi ch is encoded at rate R i . In bl ock 1, each node i t ransmits its o wn message w i (1) . At the end of block 1, each node i decodes th e messages sent by t he nod es of its 1-hop decode-set, i.e., { w j (1) : j ∈ D i (1) } . In block 2, e ach node i transmits { w i (2) , w E i (1) (1) } us ing th e binning technique, where, w E i (1) (1) stands for { w j (1) : j ∈ E i (1) } . That is, besides i ts own message w i (2) , nod e i also helps transmittin g the pre vious-block messages of the nodes in its 1-hop encode-set, which ha ve been decoded by node i si nce E i (1) ⊆ D i (1) . At the end of bl ock 2, each node i decodes the block-2 messages of the nodes in its 1-hop d ecode-set and the block-1 messages of the nodes in its 2-hop decode-set, i. e., { w D i (1) (2) , w D i (2) (1) } . Sequentially , i n block b = 3 , 4 , . . . , each node i transmit s { w i ( b ) , w E i (1) ( b − 1) , . . . , w E i ( b − 1) (1) } using the binning technique, and decodes { w D i (1) ( b ) , . . . , w D i ( b ) (1) } at the end of b lock b , where, let E i ( b ) = D i ( b ) = ∅ when b > L , and always set w ∅ ( l ) = ∅ for any l ≥ 1 . T o im plement the above omnidirectional relay scheme, we can use regular encodi ng/slidin g- window decoding wi th random binning at each node, as h as b een u sed in s e veral simple networks in [7]. Note that random binning can be replaced by deterministic binning that is easier to implement, altho ugh random binning is s impler to describe in the achiev ability proof. 5 In order to successfully carry out the above omnidi rectional relay scheme, obviously , the necessary and sufficient condit ion is that at the end of each block b = 1 , 2 , 3 , . . . , every node i ∈ N can successfully decode { w D i (1) ( b ) , . . . , w D i ( b ) (1) } . This is ess entially a multi-bl ock m ultiple- access problem , which will be d iscussed in detail i n Section IV. Apparently , the result of successfully carrying out the omni directional relay scheme for B blocks, with B ≫ L such that ( B − L ) /B ≈ 1 , i s that each nod e i recei ves the messages generated by all the nodes in the s et S L k =1 D i ( k ) , approximately at their init ial rates. Therefore, the ori ginal networking problem is solved as long as S i ⊆ S L k =1 D i ( k ) for all i ∈ N . Hence, the key step in the design of the omnidirectional relay schem e is t he selectio n o f appropriate decode-sets and encode-sets. The sizes of decode-sets are restricted by t he decoding requirement, but should be large enou gh to finall y cover all t he intended source nodes. Larger encode-sets result in more m essages being helped, but may increase t he decoding burden to some nodes that may not be interested in all the messages. It is instructive to note that finally , for any node i , the signals transmitted by all the nodes in S L k =1 D i ( k ) are decoded, eith er as useful messages, or as useless m essages but not causing interference, while the s ignals transmitted by all t he nodes i n N \ S L k =1 D i ( k ) are n ot decoded, thus causing interference. I I I . T H E A L L - S O U R C E A L L - C A S T P RO B L E M In order to demonst rate the benefit of t he om nidirectional relay scheme, in this paper , we focus on the special networking probl em where all t he no des are independent sources and each node wants t o s end it s i nformation t o all the other nodes in the network. That is, we consider the s pecial case where T i = N \{ i } for all i ∈ N , or equiv alently , S i = N \{ i } for all i ∈ N . Naturally , this can be named as the all -source all-cast p roblem. T o simplify the studies, we only address t he case where all rates R i are equal to some common rate R . W e make a very g eneral ass umption on the signal att enuation. W e only assum e that l onger distance, higher attenuation. Th at is , there is a non-increasing function to relate the m agnitude of th e gains in (1) to the distance: | g i,j | = g ( d i,j ) , (2) where d i,j is the distance between no de i and node j , and g ( · ) is some non-increasing function . For sim plicity , we assum e the s ame transmit power constraint P for all the nodes. T herefore, when a node i is transmi tting at i ts full power , the corresponding recei ved po wer at another node j i s | g i,j | 2 P . W e wi ll show that for t he all-so urce all-cast problem, it is possi ble to completely eli minate interference in arbitrarily l ar ge wireless networks, and each node can make use of the signals 6 transmitted by all the other nodes. More im portantly , we will sho w t he achie v ability of t he following common rate for the all-source all-cast problem for some network topo logies by the omnidirectional relay scheme: R < 1 n − 1 log   1 + min j P i 6 = j | g i,j | 2 P N   . (3) Obviously , P i 6 = j | g i,j | 2 P is the tot al recei ved power at no de j if the sign als transmitt ed by diffe rent nodes didn’t add up coherently at th e receiver . Thi s will be th e case if i ndependent codebooks are used at different nodes. Then, min j P i 6 = j | g i,j | 2 P corresponds to the nod e whose total recei ved power is t he least . Since every node needs to decode all the other n − 1 sources, (3) clearly is the hi ghest common rate R achiev able for the all-source all-cast problem according to the Shannon form ula. It may be poss ible to achiev e higher rates than (3) by usi ng correlated codebooks at different nodes to boost t he received power at some nodes, say , by beamforming or coherent transmi ssion. A meth od i s by using superpos ition coding as mentio ned in the Introduction. Ho we ver , thi s may be hard t o impl ement in practice due to, e.g., t he lack of channel state informat ion at t he transmitters. Moreover , note that cooperating signals mus t represent the same informati on in order to coop erate, which means that they cannot help the transm ission of other different messages. This may not be a good choi ce for t he all-source all-cast problem, where the mess ages to be transmitted by any two nodes are not comp letely the same. W e will sho w that the rate (3) is achie v able for networks wi th some kind of symmetry , which include the network depicted in Fig. 1 where the nodes are e venly s paced. In the following, we first develop a special version of t he omnidi rectional relay scheme for the all-source all-cast problem, where network topol ogy is taken into consideration. 1 2 3 Fig. 1. A reg ular network. A. A d istance-r e gulat ed omnidire ctional r elay scheme W e introdu ce the concept of k -hop neigh bors in the n etwork i n the following way . First, for each n ode i , define a set of nodes in it s neighborhoo d as its 1-hop neighbors, and denote the 7 set as N i (1) . The w ay of defining 1-hop neighbors depend s on the network topology and will be specified later on for diffe rent networks. If node j is a 1-hop n eighbor o f node i , it is said that j can reach i in one hop. If furthermore, i is a 1-hop neighbor of node l , then it is said t hat j can reach l in two hops. Similarly , it can be said that a node can reach another node in k hops, for any positive integer k . No w , for each node i , its k -hop neighbors is defined as the set of nodes that can reach it in k hops, b ut not in any less ho ps, and d enote this set as N i ( k ) . Mathematically , N i ( k ) can be sequentially d efined as N i ( k ) = { j : j ∈ N l (1) for som e l ∈ N i ( k − 1) , (4) and j / ∈ { i } ∪ N i (1) ∪ · · · ∪ N i ( k − 1) } . It is clear t hat for any network of a finite number of n odes, there is a finite n umber L i for each i ∈ N , such that N i ( k ) = ∅ for k > L i . W e use block Markov coding. In block 1, each node i transmits its own message w i (1) . At the end of block 1, each node i decodes at least the messages sent by its 1-hop neighbors { w j (1) : j ∈ N i (1) } (Maybe more can be decoded). In block 2, each node i transmits { w i (2) , w N i (1) (1) } using the binni ng techniqu e, where for simpli city , w N i (1) (1) stands for { w j (1) : j ∈ N i (1) } . At the end of block 2, each no de i decodes at l east the block-2 messages of its 1-hop neigh bors and the bl ock-1 m essages of it s 2-ho p neighbors, i.e., { w N i (1) (2) , w N i (2) (1) } . In bl ock 3 , each nod e i transm its { w i (3) , w N i (1) (2) , w N i (2) (1) } using the binning technique. Generally , in block b , each node i transmits { w i ( b ) , w N i (1) ( b − 1) , . . . , w N i ( b − 1) (1) } u sing the binning t echnique, and decodes at least { w N i (1) ( b ) , . . . , w N i ( b ) (1) } at the end of block b , where, w hen the blo ck num ber i s l ar ge enough such that N i ( b ) = ∅ , w ∅ ( l ) = ∅ for any l ≥ 1 . Obviously , E i ( k ) corresponds to N i ( k ) in t his special version, while D i ( k ) can be arbitrary as long as E i (1) ∪ · · · ∪ E i ( k ) ⊆ D i (1) ∪ · · · ∪ D i ( k ) , for any i ∈ N and k ≥ 1 . In order to solve the all-source all-cast problem where each node needs to decode the messages of all the other nodes, for t he networks to be discussed in Section V, we will choose the 1 -hop neighbor sets {N i (1) : i ∈ N } i n a way such t hat for any i ∈ N , L i [ k =1 N i ( k ) = N \{ i } . (5) T o sh ow t hat this scheme works for so me networks, we start wit h a key t echnical lemma i n next section, whi ch discusses a multi ple-access decoding based on mu ltiple blocks. 8 I V . K E Y T E C H N I C A L L E M M A : M U LT I - B L O C K M U LT I P L E - A C C E S S Consider an A WGN mult iple access channel Y ( t ) = X i ∈M X i ( t ) + Z ( t ) , (6) where, M = { 1 , 2 , . . . , m } denotes t he set of sources. According to the well known multiple-access capacity re gion [9, Ch.14], a rate v ector ( R 1 , . . . , R m ) is achiev able if and only if the i nequality X i ∈S R i < log  1 + P i ∈S P i N  (7) holds for all non-empty sub sets S ⊆ M . Namely , i f each source i ∈ M encodes its mess age w i at rate R i with independent Gaussian block code words X ¯ i ( w i ) with power P i , then (7) is the necessary and sufficient condition such that { w 1 , w 2 , . . . , w m } can be decoded, in the sense that the decodi ng error can be m ade arbit rarily small by increasing the block length. Obviously , (7) needs to hold for all nonempty S ⊆ M in order to decode { w 1 , w 2 , . . . , w m } . Howe ver , it may not be so commonly recognized that as long as (7) hol ds for the one S = M , there must b e some nonempty subset of { w 1 , w 2 , . . . , w m } that can be decoded. T his is formally stated as the foll owing lemma. Lemma 4.1: For the multi ple access channel (6), with each source i ∈ M sending a message w i at rate R i with power P i , there always exists som e nonempty subset of { w 1 , w 2 , . . . , w m } that can be decoded, as long as the following inequality hol ds: X i ∈M R i < log  1 + P i ∈M P i N  (8) i.e., (7) with S = M . Pr oof: W e use a contradiction ar gument. Suppose (7) doesn’t hold for some A ⊂ M , i. e., X i ∈A R i ≥ log  1 + P i ∈A P i N  . (9) Then t aking the dif ference between (8) and (9), we ha ve X i ∈A c R i < log  1 + P i ∈A c P i N A  (10) where, A c = M\A , and N A = P i ∈A P i + N . Now , by comparing (10) with (8), we arri ve at the same situation as (8) with M replaced by A c , and N replaced by N A . Similarly , if the inequality X i ∈S R i < log  1 + P i ∈S P i N A  (11) 9 holds for all no nempty S ⊆ A c , then t he sub set of messages { w i : i ∈ A c } can be decoded; Otherwise, if (11) doesn’t hold for som e B ⊂ A c , the process can be continued with B c = A c \B . As t he size of the subset decreases, we must be able to reach a nonempty subset where all the necessary inequali ties of the type (11) hold, and th us the messages can be decoded. This is obvious, since i f the process continues witho ut stopping, it must reach a s ubset with only one source, and by then, the sin gle inequalit y like (10) su f fices for the decodin g. Therefore, we proved th at if (8) holds, there must exist a nonempty sub set M 2 ⊆ M such that { w i : i ∈ M 2 } can be decoded, while { w i : i ∈ M 1 } with M 1 = M\M 2 cannot. Now , in our block M arkov coding settin g with relays, the no des help each other t o transfer messages. T o put int o this perspective , l et us consider a two-block decoding situ ation where in the first block { w i (1) : i ∈ M 2 } are decoded while { w i (1) : i ∈ M 1 } are not , and in the second block, each node i ∈ M 2 helps transmitting some mess ages from { w i (1) : i ∈ M 1 } besides its own mess age w i (2) . T he goal n ow is to decode { w i (2) : i ∈ M 2 } ∪ { w i (1) : i ∈ M 1 } at the end of the second block. In cons istency with our notation earlier , denote w M 1 (1) = { w i (1) : i ∈ M 1 } , w M 2 (2) = { w i (2) : i ∈ M 2 } , and { w M 2 (2) , w M 1 (1) } = { w i (2) : i ∈ M 2 } ∪ { w i (1) : i ∈ M 1 } . Denote J i ⊂ M as the set of nodes that node i helps in the second block, i.e., n ode i s ends a codew ord X ¯ i ( w i (2) , w J i (1)) by binn ing the multiple messages in the second block. Re versely , denote I i ⊂ M as the set of no des t hat will help node i to t ransmit w i (1) in the second block. For any subset S ⊆ M , let S 1 = S ∩ M 1 , and let S 2 = ( S ∩ M 2 ) ∪ ( [ i ∈S 1 I i ∩ M 2 ) . (12) That is, S 2 also consist s o f nodes from M 2 that may not be in S , but are helping transmitt ing w S 1 (1) . T hen, it can be easily verified with a ty pical sequence ar gument th at { w M 2 (2) , w M 1 (1) } can be decoded if and only if for any nonempty sub set S ⊆ M , X i ∈S R i < log  1 + P i ∈S 1 P i N  + log  1 + P i ∈S 2 P i P i ∈M 1 P i + N  (13) where the first term is the contribution of the nodes in S 1 from the first block, and the s econd term is the cont ribution of th e nodes in S 2 from the second block. Actually , it is rather in structive to think of the constraints (13) for all nonempty S ⊆ M as a two-block multiple-access re gion. Although it is necessary that the in equality (13) should hold for all nonempt y S ⊆ M in order to decode { w M 2 (2) , w M 1 (1) } , as i n t he case o f one-block mu ltiple-access discussed earlier , we will show that t he fol lowing si ngle inequalit y X i ∈M R i < log  1 + P i ∈M 1 P i N  + log  1 + P i ∈M 2 P i P i ∈M 1 P i + N  (14) 10 i.e., (13) with S = M , is enough t o ensu re that some nonemp ty s ubset of { w M 2 (2) , w M 1 (1) } can be decoded. W e still use a contradiction ar g ument. If (13) ho lds for all nonempty S ⊆ M , then { w M 2 (2) , w M 1 (1) } can be decoded; Ot herwise, if for some nonempty A ⊂ M , (13) doesn’t hold, i.e., X i ∈A R i ≥ log  1 + P i ∈A 1 P i N  + log  1 + P i ∈A 2 P i P i ∈M 1 P i + N  (15) then t aking the dif ference between (14) and (15), we h a ve X i ∈A c R i < log 1 + P i ∈A c 1 P i P i ∈A 1 P i + N ! + log 1 + P i ∈A c 2 P i P i ∈A 2 P i + P i ∈M 1 P i + N ! (16) where, A c = M\A , A c 1 = M 1 \A 1 , and A c 2 = M 2 \A 2 . By the definition (12), it si mply foll ows that A ⊆ A 1 ∪ A 2 and A c ⊇ A c 1 ∪ A c 2 . Hence, by replacing A c with A c 1 ∪ A c 2 in the left-hand-side of (16), we have X i ∈A c 1 ∪A c 2 R i < log 1 + P i ∈A c 1 P i P i ∈A 1 P i + N ! + log 1 + P i ∈A c 2 P i P i ∈A 2 P i + P i ∈M 1 P i + N ! . (17) This is the same sit uation as (14) wit h M replaced by A c 1 ∪ A c 2 , M 1 replaced by A c 1 , M 2 replaced by A c 2 , and some adjustm ent of the noises. Now , the messages to be decoded are { w A c 2 (2) , w A c 1 (1) } . As in the case of one-block multiple-access discussed earlier , such a process can be continued unt il we find a nonempty subset of { w M 2 (2) , w M 1 (1) } that can be decoded. Therefore, we proved that the inequalit y (14) alone ensures t hat there always exists a nonempt y subset of { w M 2 (2) , w M 1 (1) } that can be decoded. Note that by combining the two terms o n the right-hand-side, (14) becomes X i ∈M R i < log  1 + P i ∈M P i N  (18) which i s exactly the same as (8). In other words, the inequality (8) or (18) makes sure th at there are always some messages that can be d ecoded, no mat ter whether it is one-block mul tiple-access, or two-block multiple access wi th relays. It is now clear that g enerally we have the following conclusion for K -block mul tiple-access with relays. Lemma 4.2: Consid er a K -bl ock decodin g situation where { w M K ( K ) , . . . , w M 1 (1) } are to be decoded for some disjoi nt subsets M k , k = 1 , . . . , K wi th S K k =1 M k = M , or equiv alently to say , th at { w M k ( k − 1) , . . . , w M k (1) } have been decoded for k = 2 , . . . , K . D uring each block k = 2 , . . . , K , every node i ∈ M k helps transmitti ng a sub set of { w M k − 1 ( k − 1) , . . . , w M 1 (1) } besides it s own message w i ( k ) with the binning techniqu e. Then there i s always a nonempty subset of { w M K ( K ) , . . . , w M 1 (1) } that can be decoded i f (18) holds. 11 V . N E T W O R K S W I T H S Y M M E T R I C T R A FF I C After the discus sion of last section, it is clear that no m atter ho w complicated the relay si tuation is, at the end of each block b , e very node i can always decode the new messages w ( b ) of a nonempty set of nod es G i ( b ) ⊂ N , under t he conditio n (3). (More detailed ar guments about this wil l be presented in the proof of Th eorem 5.1). Then in order t o s uccessfully carry out t he distance-regulated omnidirectional relay scheme presented in Section III, we only need to make sure th at for each i ∈ N N i (1) ⊆ G i ( b ) , for all b = 1 , . . . , B . (19) Due to t he monot onicity of t he power att enuation model (2), messages s ent by nodes that are closer are generally easier to decode, and t herefore it is natural to choose N i (1) as a set composed of the clos est nodes. In vi e w of the requirement (19), it is prefer able to put as fe w as possible nodes into N i (1) . Howe ver , for the all-source all-cast problem, each N i (1) should contain suffi ciently many nod es so that (5) holds, i.e., the whole net work wil l be cov ered and each node will decode the m essages of all the other nodes. When we are s ure that there are some nodes whose mess ages can be decoded but not knowing how many of them th ere are, it is not clear whether the messages of all the nodes in N i (1) can be decoded if N i (1) contains more than one nodes. Howe ver , there is a special situation where we can be sure, i. e., when there is some ki nd of symmetry to all the n odes in N i (1) , in the sens e that if one of them can be decoded, the others certainly can. T wo sim ple examples of this are sho wn in Fig. 2 , where clearly , for any node i , the t raf fic is symmetric on both si des. For each node i , by cho osing N i (1) as it s two n eighboring nodes, which are most ly easy to decode, it is certain that b oth of them can be decoded. Sin ce (5) obviously holds by this definition, the all-source all-cast pro blem for these networks is solved under the condition (3). Fig. 2. T wo symmetric networks. 12 The network depicted in Fig. 1 is not completely sy mmetric. For any no de not in the center , i.e., i 6 = ( n + 1) / 2 , th e traf fic on one sid e is hea vier than the other side. Ho wev er , there is still some k ind of sy mmetry as we will show later on, s o th at an y non -boundary n ode i / ∈ { 1 , n } can decode the messages o f both its neighbors { i − 1 , i + 1 } simultaneously , under the condition (3). W e will first prove this for more general network topo logies, and then the regular topology in Fig. 1 will follow as a simpl e corollary . Alternatively , a direct proof for the regular network i n Fig. 1 has been presented in [10]. 1 2 3 Fig. 3. A general one-dimensional network. Consider a general wi reless network of n nodes lo cated on a st raight line, labeled sequent ially by 1 , 2 , . . . , n , as depicted in Fig. 3. It is con venient to introduce th e not ation P i,j = | g i,j | 2 P for any i, j ∈ { 1 , 2 , . . . , n } . For any node i / ∈ { 1 , n } , let its 1-hop neighb ors be N i (1) = { i − 1 , i + 1 } . Let N 1(1) = { 2 } and N n (1) = { n − 1 } . W e wi ll show that the distance-regulated omn idirectional relay scheme presented i n Section III works for this one-dimensional network as long as the common rate R satisfies (3) and the following two symmetric sets of constraints for ever y i ∈ { 2 , 3 , . . . , n − 1 } : i) For any ℓ = 1 , . . . , i − 2 , at least one of the following two inequalities holds: ( ℓ + n − i ) R < log  1 + P 1 ,i + · · · + P ℓ,i + P i +1 ,i + · · · + P n,i N  (20) or ( n − i ) R < log  1 + P i +1 ,i + · · · + P n,i P 1 ,i + · · · + P ℓ,i + N  (21) ii) For any r = i + 2 , . . . , n , at least on e of t he foll owing two inequalities holds: ( i + n − r ) R < log  1 + P 1 ,i + · · · + P i − 1 ,i + P r,i + · · · + P n,i N  (22) or ( i − 1) R < log  1 + P 1 ,i + · · · + P i − 1 ,i P r,i + · · · + P n,i + N  (23) In ot her words, we hav e the fol lowing theorem. Theor em 5.1: For the one-dimensio nal wireless network, a common rate R is achie vable for the all-source all-cast problem with the omnidirectional relay s cheme, if it satisfies (3), and also for every i ∈ { 2 , 3 , . . . , n − 1 } , the above constraints i) and ii) hold. 13 Pr oof: W ith t he distance-regulated o mnidirectional relay scheme, we only need to s how th at at th e end of each bl ock b , ev ery node i can decode w N i (1) ( b ) . Obviously , for any node i and at the end of any block b , there are a sequence of disj oint subsets M k , k = 1 , . . . , b (can b e empty) wit h S b k =1 M k = N \{ i } such that { w M b ( b ) , w M b − 1 ( b − 1) , . . . , w M 1 (1) } are to be decoded, or equi v alently to say , that { w M k ( k − 1) , . . . , w M k (1) } h a ve been decoded for any k = 2 , . . . , b i n t he previous blocks. Then according to Lemma 4.2, there must e xist a nonempt y sub set of { w M b ( b ) , w M b − 1 ( b − 1) , . . . , w M 1 (1) } that can be decoded, due to ( n − 1) R < log  1 + P 1 ,i + · · · + P i − 1 ,i + P i +1 ,i + · · · + P n,i N  (24) which follows from (3). If t his no nempty subset is disjoi nt wit h w M b ( b ) , then after t he d ecoding, we arriv e at a simil ar situation with another sequence of dis joint M ′ k , k = 1 , . . . , b wi th S b k =1 M k = N \ { i } . Then Lemma 4.2 can be applied again with (24) so that m ore messages can be decoded. This process can be continued as long as all nodes i n N \{ i } hav e messages to be decoded. In other words, finally , there must be a nonempty s ubset M ∗ b ⊆ N \{ i } such that w M ∗ b ( b ) can be decoded at th e end of block b . According to the relay structure a nd the monotoni city of th e po wer attenuation, M ∗ b can only be one of the following t hree types of subsets of no des: { ℓ, . . . , i − 1 } for some ℓ < i ; { i + 1 , . . . , r } for s ome r > i ; o r { ℓ, . . . , i − 1 , i + 1 , . . . , r } for som e ℓ < i < r . This is simp ly based on the ob serva tion that on either side, it is alw ays easi er to decode m essages from nodes cl oser . If i is a b oundary n ode, i.e., 1 or n , then onl y one of the first t wo types i s po ssible and clearly N i (1) ⊂ M ∗ b . Now , for a non-boun dary node i ∈ { 2 , 3 , . . . , n − 1 } , all three types are poss ible. If M ∗ b is of the third type, then clearly , N i (1) ⊂ M ∗ b and the proof is finished. If M ∗ b is of the first type, then Lemma 4.2 stil l can be applied with either (20) or (21) continually until w i +1 ( b ) is decoded. Note that the case (21 ) is different from (20) in t he sense that there is no intension to decode the m essages of the no des { 1 , . . . , ℓ } , and their transm issions are t reated as nois e. Actually , they m ay not be all causing i nterference in all blocks , and hence, the condition needed to apply L emma 4.2 may be weaker than (21). Symmetrically , it can be sho wn that w i − 1 ( b ) will be decoded b ased on either (22) or (23 ) if M ∗ b is of the second type. Therefore, we’ ve s hown that w N i (1) ( b ) wil l always be decoded. This conclud es the proof. Now , we show that for t he regular network in Fig. 1, any rate satisfying (3) must satis fy the constraints i ) and ii). Thus, t he rate (3) is achiev able. Due to the equal separation dist ance d 0 and the power gain model (2), it is con venient to define P i = g ( id 0 ) P for any i ≥ 1 . 14 Then, P i,j = P | i − j | for any i 6 = j . According to the monotonicit y of the function g ( · ) , we have P 1 ≥ P 2 ≥ · · · ≥ P n − 1 . (25) W i th this ne w notatio n, (3) becomes R < 1 n − 1 log  1 + P 1 + P 2 + . . . + P n − 1 N  (26) where, t he t otal received power corresponds to any one of the boundary nodes, and is the sm allest among all the nodes. The constraint s i) and ii) become: For ev ery i ∈ { 2 , 3 , . . . , n − 1 } , i) For any ℓ = 1 , . . . , i − 2 , at least one of the following two inequalities holds: ( ℓ + n − i ) R < log 1 + P i − 1 j = i − ℓ P j + P n − i j =1 P j N ! (27) or ( n − i ) R < log 1 + P n − i j =1 P j P i − 1 j = i − ℓ P j + N ! (28) ii) For any r = i + 2 , . . . , n , at least on e of t he foll owing two inequalities holds: ( i + n − r ) R < log 1 + P i − 1 j =1 P j + P n − i j = r − i P j N ! (29) or ( i − 1) R < log 1 + P i − 1 j =1 P j P n − i j = r − i P j + N ! (30) Now , we verify that at least on e of (27 ) and (28) must ho ld. First, note that by the conca vity of th e logarit hmic function, it follows from (25) and (26) t hat for any 1 ≤ k ≤ n − 1 , k R < log  1 + P 1 + P 2 + . . . + P k N  . (31) Specially , when k = i − 1 , we have ( i − 1) R < log  1 + P 1 + P 2 + . . . + P i − 1 N  . (32) If i − ℓ ≤ n − i + 1 , by (25), we hav e i − 1 X j = i − ℓ P j + n − i X j =1 P j ≥ n − i + ℓ X j =1 P j and th us, by (31) with k = ℓ + n − i , (27) hold s. Otherwise, if i − ℓ > n − i + 1 , we check the following inequality ℓR < log 1 + P i − 1 j = i − ℓ P j N ! . (33) 15 If (33) holds, t hen by (32), (25) and the concavity of the logarith mic functi on, (27) follows. Otherwise, if (33) doesn ’ t hold , i.e., ℓR ≥ log 1 + P i − 1 j = i − ℓ P j N ! , (34) taking the differe nce betw een (32) and (34), we have ( i − 1 − ℓ ) R < log 1 + P i − ℓ − 1 j =1 P j P i − 1 j = i − ℓ P j + N ! . (35) Then again by (25) and the conca vity of the logarith mic function, we hav e (28). Similarly , by symmetry , we can sho w that at least one of (29 ) and (30) must hold. Therefore, we arrive at the following theorem. Theor em 5.2: For the one-dimensi onal regular wireless network in Fig. 1 , the common rate (26) is achie vable for the all-source all-cast p roblem with the om nidirectional relay schem e. 1 2 3 4 5 6 7 8 9 Fig. 4. A general network with nodes clearly ordered by distance. 1 2 3 Fig. 5. A reg ular network wi th a clear ordering of nodes by distance. Remark 5.1: In all the ar guments above, obviously , it is not necessary for all the nodes to be located on a straight line, as long as they can be clearly ordered in terms of the dist ances, i.e., there is a way of labelin g t he nodes so that d i,j ≤ d i,k for any i < j < k , or k < j < i . One such example is s hown in Fig. 4, and a regular case is shown in Fig. 5. In such cases, Theorem 5.1 or 5.2 st ill applies. V I . C O N C L U S I O N W e d e veloped an omni directional relay scheme for wireless networks with mul tiple sou rces, where each node can simult aneously relay mult iple messages in di ff erent directions by binning 16 them into a single si gnal. T his scheme also exploits the broadcast n ature of wireless commu- nication, such that one node helps mul tiple n odes, and multiple nodes help one no de. In the extreme, this scheme is capable of completely elimin ating interference in the whole network, and sp ecially , for the all-source all-cast problem where all m essages are of interest, each node can b enefit from the sig nals transmit ted by all the other nodes. W e als o dem onstrated some kind of opt imality of thi s scheme by showing that it achieves the m aximum rate p ossible for s ome networks if no beamforming i s performed. W e proposed a di stance-regulated networking frame work, which was shown t o work well for some networks. 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