A Greedy Omnidirectional Relay Scheme

1 A Greedy Omnidirect ional Relay Scheme Liang-Liang Xie Departmen t of E lectrical an d Com puter En gineerin g University of W aterloo, W aterloo, ON, Canada N2L 3G1 Email: llxie@ece. uwaterloo.ca Abstract A greedy om nidirectio nal relay sche me is d ev eloped, and the co rrespon ding achievable rate region is obtained for the all-source all-cast prob lem. The discussions are first b ased on the genera l discrete memory less channel model, a nd then applied to the a dditive white Gaussian noise (A WGN) models, with both full-duplex and h alf-dup lex modes. I . I N T RO D U C T I O N A general framework of omni directional relay has been deve loped in [1]-[4]. It generalizes the decode-and-forward re lay st rategy introduced in [5] with the netw ork coding idea introduced in [6] to the case of wireless networks wit h multiple sources. T echnically , it is a comb ination of block Markov coding with binni ng, s o that each relay can simultaneousl y transport mult iple messages i n dif ferent di rections. The ef fecti veness of this o mnidirectional relay strategy has been demonstrated by the result that it is poss ible to completely elimi nate interference in t he network, and each node can ful ly exploit the signals transmitted by all the ot her nodes. In this paper , we dev elop a special “greedy” omnidirectional relay scheme in the sense that each node tries t o relay as many messages as possible. W ithout being regulated by network topolo gies, this greedy scheme is s imple to implement, and can be adaptive to time-v arying situations. Our discuss ion will first be o n the general discrete memoryless channel model. And then, motiv ated by wireless networks, the results will be appl ied to the A WGN models, with both full- duplex and half-dupl ex modes. For simplicity , in t his paper , we focus on the all-source all-cast problem, and obtain a general achiev abl e rate region. I I . A G E N E R A L D I S C R E T E M E M O RY L E S S N E T W O R K C H A N N E L M O D E L Consider a network of n n odes N = { 1 , 2 , . . . , n } , wit h the channel modeled by ( X 1 × · · · × X n , p ( y 1 , . . . , y n | x 1 , . . . , x n ) , Y 1 × · · · × Y n ) . At each time t = 1 , 2 , . . . , every node i ∈ N sends an input X i ( t ) ∈ X i , and receiv es an output Y i ( t ) ∈ Y i , and they are related vi a p ( Y 1 ( t ) , . . . , Y n ( t ) | X 1 ( t ) , . . . , X n ( t )) . 2 I I I . A G R E E D Y 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 The essence of this “greedy” scheme is that at th e end of each block, ev ery node decodes as many messages as pos sible, and in th e next block, relays all th e messages it has decoded, with the re striction of adding at most one ne w message for ea ch source. T o be m ore specific, e very node i relays the message w j ( b 0 ) , if it has decoded it, and it has relayed all the messages w j ( b ) , b = 1 , . . . , b 0 − 1 previously . Consider the all-source all-cast problem, where each node i is an independent source, and wants to send som e common information to all the other nodes at the rate R i . W ith this greedy omnidirectional relay s cheme, we hav e the following achie v able rate region for the all-source all-cast probl em. Theor em 3.1: Consider th e al l-source all-cast problem. W ith the greedy omnidirectional relay scheme, a rate vector ( R 1 , R 2 , . . . , R n ) is achiev able if for an y nonempty subset S ⊂ N , there is a node i 0 ∈ S , such that X j ∈S c R j < I ( X S c ; Y i 0 | X S ) (1) for som e p ( x 1 ) p ( x 2 ) · · · p ( x n ) , where X S c = { X j : j ∈ S c } , and X S = { X i : i ∈ S } . For three-node networks, the achie v ability of the rate region prescribed by (1) has been p roved in [2, Thm 4.1], where , instead of th e greedy relay scheme, the relay ordering was set according to the relativ e s trengths of the chann els between different nodes. Howe ver , even for three-node networks, the proof in [2] turned out to be rath er com plicated, since there were too many different cases to address. Here, in Section VI of this paper , we will present a simple and general proof based on the greedy relay scheme, which applies to networks wit h any num ber of nodes. Now , we consider a tim e-va rying operation of t he n etwork, with diff erent input distributions in diffe rent blocks. Specially , we are interested in the period ic case, where the input dis tribution in block b is p k ( x 1 ) p k ( x 2 ) · · · p k ( x n ) with k = ( b mod K ) for som e period K ≥ 2 . Correspondingl y , we have the following conclusion. Theor em 3.2: Consider the all-source all-cast problem. W ith the periodic greedy omnidirec- tional relay scheme, a rate vector ( R 1 , R 2 , . . . , R n ) is achie va ble if for any nonempty subset S ⊂ N , there is a node i 0 ∈ S , such that X j ∈S c R j < 1 K K X k =1 I k ( X S c ; Y i 0 | X S ) where, the mutual information I k is calculat ed based on p k ( x 1 ) p k ( x 2 ) · · · p k ( x n ) . Obviously , to obtain more g eneral results, we can also consider dif ferent block lengths. Let block b ha ve length L k with k = ( b mod K ) . Then, we hav e the foll owing conclusion. 3 Theor em 3.3: Consider the all-source all-cast problem. W ith the periodic greedy omnidirec- tional r elay scheme with v arying block lengths, a rate vector ( R 1 , R 2 , . . . , R n ) is achie va ble if for any nonempty subset S ⊂ N , there is a node i 0 ∈ S , such that X j ∈S c R j < 1 P K k =1 L k K X k =1 L k I k ( X S c ; Y i 0 | X S ) where, the mutual information I k ( · ) i s calculated based on p k ( x 1 ) p k ( x 2 ) · · · p k ( x n ) . I V . F U L L - D U P L E X A W G N W I R E L E S S N E T WO R K S Consider the follo wing A WGN wireless network channel model with full-duplex mode: Y j ( t ) = X i ∈N i 6 = j g i,j X i ( t ) + Z j ( t ) , ∀ j ∈ N , t = 1 , 2 , . . . (2) where, X i ( t ) ∈ C 1 and Y i ( t ) ∈ C 1 respectiv ely d enote the signals sent and rec eiv ed by Node i ∈ N at time t ; { g i,j ∈ C 1 : i 6 = j } denote the signal attenuation gains; and Z i ( t ) is zero-mean complex Gaussian noi se with variance N . Consider the a ver age power constraint: 1 T T X t =1 | X i ( t ) | 2 ≤ P i for all T = 1 , 2 , . . . , and i ∈ N . Then appl ying Theorem 3 .1, we h a ve the following conclusion. Theor em 4.1: Consider the all -source all -cast problem for the full-du plex A WGN wireless net- works. W ith the greedy omnid irectional relay scheme, a rate vector ( R 1 , R 2 , . . . , R n ) is achie vable if for any nonempty subset S ⊂ N , there is a node i 0 ∈ S , such that X j ∈S c R j < log 1 + P j ∈S c | g j,i 0 | 2 P j N ! . V . H A L F - D U P L E X A W G N W I R E L E S S N E T W O R K S Consider the foll owing A WGN wireless network channel model with half-dupl ex mode: At time t = 1 , 2 , . . . , the transmitter set is T ( t ) ⊂ N , and the receiv er set is R ( t ) = N \T ( t ) , and Y j ( t ) = X i ∈T ( t ) g i,j X i ( t ) + Z j ( t ) , ∀ j ∈ R ( t ) , (3) where, X i ( t ) ∈ C 1 and Y j ( t ) ∈ C 1 respectiv ely denote the sign al sent by node i and the signal recei ved by node j at time t ; { g i,j ∈ C 1 : i 6 = j } denote the s ignal attenuation gain s; and Z j ( t ) is zero-mean complex Gaussian noise with v ariance N . 4 Consider the follo wing a verage power constraint: P T t =1 | X i ( t ) | 2 I [ i ∈T ( t )] P T t =1 I [ i ∈T ( t )] ≤ P i for all T = 1 , 2 , . . . , and i ∈ N , where I [ · ] is the indicator function: I [ i ∈T ( t )] = ( 1 , if i ∈ T ( t ) , 0 , otherwise. Consider a periodically b lock-varying o peration of the network. In block b = 1 , 2 , . . . , the block length is L k , the transmi tter set is T k , and t he rec eiv er set is R k , with k = ( b m od K ) for some period K ≥ 2 . Then by Theorem 3.3, we ha ve the following conclusion. Theor em 5.1: Consider the all-source all-cast problem for the half-duplex A WGN wireless networks. W ith the periodic greedy omnidirectional relay scheme with varying b lock l engths, a rate vector ( R 1 , R 2 , . . . , R n ) is achie va ble if for any n onempty sub set S ⊂ N , there is a node i 0 ∈ S , such that X j ∈S c R j < 1 P K k =1 L k K X k =1 L k I [ i 0 ∈R k ] log 1 + P j ∈S c ∩T k | g j,i 0 | 2 P j N ! . V I . P RO O F O F T H E T H E O RE M S Pr oof of Theor em 3.1: The key to the proof is the technical Lemma 4.1 de veloped in [4], which basi cally says that once the inequality (1) holds, n ode i 0 can always decode the messages of some non empty subset of S c . W e wi ll prove by induction that each n ode can decode the messages sent by all the other nodes. According to the greedy relay scheme, once a node i has d ecoded some mess ages of another node j , it wi ll al ways transmit t he messages of n ode j in the subsequent blocks. W e say that node i cov ers a set of nodes S , if node i has decoded some messages of e very node in S , and therefore, will transm it the messages of every node in S in the subsequent blocks. It is obvious that at t he end of any bl ock b ≥ 1 , ea ch node i can decode the block- b transm ission of some other node j i 6 = i , by applyin g the Lemma to (1) with S = { i } . In other w ords, at the end of block b , each node i will at least cover what h a ve been covered by certain two nodes { j i , i } at the end of block b − 1 . For b ≥ 2 , since at the end of block b − 1 , each one of the two nodes { j i , i } m ust ha ve covered what had been co vere d by at least a pair of nodes at the end o f block b − 2 , we have that at the end of block b , nod e i will at least co ver what had been cover ed by three nodes at the end of block b − 2 . T o see th is, there are two cases: If at least one of the two pairs is dif ferent from { j i , i } , the total covering is obviously at l east three nodes; If both the 5 two pairs are identi cal to { j i , i } , then one of the two nodes { j i , i } must be able t o cover anot her node according to the Lemma applied to (1) wit h S set to { j i , i } , thus stil l leading to a covering of at least three nodes. Therefore, at the end of any block b ≥ 2 , each node wil l at least cov er what had been covere d by certain three nod es at t he end of block b − 2 . Now , since at t he end of any block b ≥ 4 , each node i at least cove rs what had been covered by certain three nodes { j i , k i , i } at the end of block b − 2 , while each o f them in turn m ust ha v e cove red what had been covere d by a set of three nodes at the end of block b − 4 , we have that at the end of bl ock b , node i wi ll at least cover what had been cov ered by four nodes at th e end of b lock b − 4 . T o see this, similarly there are two cases: If at least one of the three sets is diffe rent from { j i , k i , i } , the tot al covering is at least four nodes; If all the three sets are i dentical to { j i , k i , i } , t hen one of the t hree nodes { j i , k i , i } must be able to cover anoth er node according to the Lemma applied to (1) with S set to { j i , k i , i } , th us s till leading to a covering of at least four nodes. Therefore, at the end of any block b ≥ 4 , each node wi ll at least co ver what had been covered by certain four nodes at the end o f block b − 4 . Inductive ly , it is easy to see that at the end o f an y block b ≥ 2 m − 2 , e ach node will at least cove r what had been co vered by certain m nodes at the end of block b − 2 m − 2 . Since eac h node cove rs its elf by the end of blo ck 0, for a network of any finite n nodes, each node wil l cover the whole network, i.e., be able to decode some messages of any o f the other nodes, at least by the end of block b = 2 n − 2 . Before we conclude the proof, we need to demonstrate that the decoding delay is finite, so that there is no rate loss by block Markov coding. W e use a contradiction ar gument. Suppose that the delay of some node i decoding t he messages of another node j is not upper bo unded, i.e., lim sup b →∞ [ D i ( w j ( b )) − b ] = ∞ (4) where D i ( w j ( b )) d enotes th e block, by the end of which, node i decodes t he message w j ( b ) —the block- b message of node j . Since at the end of any block b ≥ 1 , node i always decodes the block- b transmissio n of another node, if (4) holds, then there must e xist another node i 1 6 = i , such th at lim sup b →∞ [ D i 1 ( w j ( b )) − b ] = ∞ . (5) In f act, if no other nodes encounter an unbounded delay , then no other nodes will relay w j ( b ) with an unbounded delay , and then node i will not decode w j ( b ) with an unbounded delay . Now , si nce both i and i 1 encounter unbounded delay in decoding w j ( b ) , for the same reason as above, there m ust be a third node that encount ers unbounded delay in decoding w j ( b ) . This 6 ar gument can be conti nued, so that all th e no des hav e to encounter unbound ed delay in decoding w j ( b ) , includ ing no de j itself. T his is obviously in contradiction. Therefore, (4) cannot hold, i.e., all th e decoding delays in the network must be uniformly bounded by s ome cons tant T 0 .  Proofs o f Th eorems 3.2 and 3.3 follow similarly by treating ev ery K bl ocks tog ether as a group block, and applying the ar gument abo ve to the group blocks. Theorems 4.1 and 5.1 are simple appl ications. R E F E R E N C E S [1] L.-L. Xie, “Network coding and random binning for multi-user channe ls, ” in vited talk at the 2007 IEEE Communication Theory W orkshop , (Sedona, Arizona), May 20 07. [2] L.-L. Xie, “Netwo rk coding and random binning for multi-user channels, ” in P r oc. IEEE Cana dian W orksho p on Information Theory , (Edmonton, Canada ), June 2007. [3] L.-L. Xie, “Omnidirectional relay in wireless networks, ” in Proc. 200 8 IEEE International Symposium on Information Theory , (T oronto, Cana da), July 2008. [4] L.-L. Xie, “Omnidirectional relay in wireless netw orks, ” submitted to IEEE Tr ans. Info rm. Theory , No vemb er 2008. [5] T . Cover and A. E l Gamal, “Capacity theorems for the relay ch annel, ” IEEE T rans. Inform. Theory , v ol. 25, pp. 5 72–584 , 1979. [6] R. Ahlswede, N. Cai, S.-Y . R. L i, and R. W . Y eung, “Network information flow , ” IEEE T rans. Inform. Theory , vol. 46, pp. 1204– 1216, July 2000.