A Greedy Omnidirectional Relay Scheme

A general framework of omnidirectional relay has been developed in [1]- [4]. It generalizes the decode-and-forward relay strategy introduced in [5] with the network coding idea introduced in [6] to the case of wireless networks with multiple sources. Technically, it is a combination of block Markov coding with binning, so that each relay can simultaneously transport multiple messages in different directions. The effectiveness of this omnidirectional relay strategy has been demonstrated by the result that it is possible to completely eliminate interference in the network, and each node can fully exploit the signals transmitted by all the other nodes.

In this paper, we develop a special “greedy” omnidirectional relay scheme in the sense that each node tries to relay as many messages as possible. Without being regulated by network topologies, this greedy scheme is simple to implement, and can be adaptive to time-varying situations.

Our discussion will first be on the general discrete memoryless channel model. And then, motivated by wireless networks, the results will be applied to the AWGN models, with both fullduplex and half-duplex modes. For simplicity, in this paper, we focus on the all-source all-cast problem, and obtain a general achievable rate region.

Consider a network of n nodes N = {1, 2, . . . , n}, with the channel modeled by

At each time t = 1, 2, . . ., every node i ∈ N sends an input X i (t) ∈ X i , and receives an output

The essence of this “greedy” scheme is that at the end of each block, every node decodes as many messages as possible, and in the next block, relays all the messages it has decoded, with the restriction of adding at most one new message for each source. To be more specific, every node i relays the message w j (b 0 ), if it has decoded it, and it has relayed all the messages

Consider the all-source all-cast problem, where each node i is an independent source, and wants to send some common information to all the other nodes at the rate R i . With this greedy omnidirectional relay scheme, we have the following achievable rate region for the all-source all-cast problem. Theorem 3.1: Consider the all-source all-cast problem. With the greedy omnidirectional relay scheme, a rate vector (R 1 , R 2 , . . . , R n ) is achievable if for any nonempty subset S ⊂ N , there is a node i 0 ∈ S, such that

for some 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 achievability of the rate region prescribed by (1) has been proved in [2,Thm 4.1], where, instead of the greedy relay scheme, the relay ordering was set according to the relative strengths of the channels between different nodes. However, even for three-node networks, the proof in [2] turned out to be rather complicated, 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 with any number of nodes. Now, we consider a time-varying operation of the network, with different input distributions in different blocks. Specially, we are interested in the periodic case, where the input distribution in block b is

we have the following conclusion. Theorem 3.2: Consider the all-source all-cast problem. With the periodic greedy omnidirectional relay scheme, a rate vector (R 1 , R 2 , . . . , R n ) is achievable if for any nonempty subset S ⊂ N , there is a node i 0 ∈ S, such that

where, the mutual information

Obviously, to obtain more general results, we can also consider different block lengths. Let block b have length L k with k = (b mod K). Then, we have the following conclusion.

Theorem 3.3: Consider the all-source all-cast problem. With the periodic greedy omnidirectional relay scheme with varying block lengths, a rate vector (R 1 , R 2 , . . . , R n ) is achievable if for any nonempty subset S ⊂ N , there is a node i 0 ∈ S, such that

where, the mutual information

Consider the following AWGN wireless network channel model with full-duplex mode:

where, X i (t) ∈ C 1 and Y i (t) ∈ C 1 respectively denote the signals sent and received by Node i ∈ N at time t; {g i,j ∈ C 1 : i = j} denote the signal attenuation gains; and Z i (t) is zero-mean complex Gaussian noise with variance N.

Consider the average power constraint:

|X i (t)| 2 ≤ P i for all T = 1, 2, . . . , and i ∈ N .

Then applying Theorem 3.1, we have the following conclusion. Theorem 4.1: Consider the all-source all-cast problem for the full-duplex AWGN wireless networks. With the greedy omnidirectional relay scheme, a rate vector (R 1 , R 2 , . . . , R n ) is achievable if for any nonempty subset S ⊂ N , there is a node i 0 ∈ S, such that

Consider the following AWGN wireless network channel model with half-duplex mode: At time t = 1, 2, . . ., the transmitter set is T (t) ⊂ N , and the receiver set is R(t) = N \T (t), and

where, X i (t) ∈ C 1 and Y j (t) ∈ C 1 respect

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