We have obtained the capacity for a class of wireless erasure networks with broadcast and no interference at reception. We have generalized some of the capacity results that hold for wireline networks [9],[54] to these networks. Furthermore, we have shown that linear encoding suffices to achieve the optimal performance. We see from the proof that it is not necessary to perform channel coding and network coding separately from each other. In fact, as we will see in Chapter 4, decoding at the relay nodes and operating below the capacities of each link can actually significantly reduce the achievable rate (for more examples, see [27],[55]). Therefore, unlike the wireline
scenario where each link is made error free by channel coding and network coding is then employed on top of that, our scheme only requires a single encoding function. Only the destination has to decode the received signal.
Many problems related to wireless networks remain open. Generalizing the results in this work for other network problems is one possible extension of this work. It will also be interesting to see if similar results can be obtained for other types of networks, such as erasure wireless networks, in which interference is incorporated in the reception model, networks involving channels other than erasure channels, etc.
Chapter 3
Broadcast Problems over Wireless
Networks
3.1
Introduction
Different traffic patterns are present in today’s communication networks. In addition to pairwise communications in the network, it is possible that a collection of users are interested in the same type of information generated in the network. This scenario resembles the multicast problem that was considered in Chapter 2 for a specific model of wireless networks. Another possibility is that a collection of users demand different types of information from a specific user referred to as source node. We refer to this problem as the broadcast problem in the network. What is apparent from the above two scenarios is that unlike point-to-point communication systems, where the source (transmitter) and the sink (receiver) of the information are specified, in communica- tion networks a user can be a transmitter and receiver (or even relay) for different types of information at the same time, and this gives rise to different network prob- lems. At a high level of generality, a network problem over a communication network is specified by the set of information messages, the source nodes, and the destina- tion nodes. Each source node has access to a subset of information messages, and each destination demands some subset of the information messages. In Section 2.3 of Chapter 2 we defined a network problem, P by a quintuple P = (M,S,D,Υg,Υr),
whereMis the set of information message,S andDare the set of source and destina- tion nodes, and Υr (respectively Υg) is the function that specifies the set of messages requested (generated) at each destination (source).
Note that the above definition is independent of the communication model defined for the network. A network problem can be defined over wireline and wireless, fixed, or mobile networks. Given a communication model for the network (which specifies the connectivity of the users and their communication capabilities), the main questions regarding a network problem P are
• What is the set of possible rates (for information messages) that can be sup- ported for P over the network?
• What is the optimal communication strategy to achieve those rate?
As mentioned in Chapter 2, these questions are answered for multicast problems over wireline networks in [9, 10] and a class of deterministic networks in [32]. We also looked at multicast problems over a class of wireless networks called wireless erasure networks and found the capacity region and the optimal strategy in Chapter 2. However, for a general network problem, the answer to the above questions is unknown. In [33, 23], outer and inner bounds on the capacity region of general network problems over wireline networks are obtained.
In this chapter we considerbroadcast problems. In a broadcast problem one source has access to multiple information messages. Each of these information messages is requested by a particular destination. In accordance to our definition, a broadcast
problem Pbc= (M,S,D,Υg,Υr) is a network problem whereS ={s}, Υg =M, and
Υr(.) is a partition ofM. There are many scenarios that can be modeled by broadcast problems. A well-known example is TV broadcasting stations. Downlink of cellular systems where a base station provides service to different users is another example of such problems that has received a lot of attention in the past few years. It should be
noted that the capacity region of a broadcast problem for a general network is still unknown. In fact, even in the multi-user setup where there are no relays present in the network and the destinations are directly connected to the source, the broadcast problem is not completely solved. In this case the problem is referred to asbroadcast
channels, which was first introduced by Cover in [13].
In a network setup the broadcast problem is solved for wireline networks with error-free links in [54]. The capacity region in this case has a min-cut interpretation. Furthermore, it can be shown that in this case the capacity is achieved by routing, and no coding at the intermediate nodes is required.
In this chapter we look at broadcast problems over wireless erasure networks. These networks were introduced in Chapter 2. At the beginning of the chapter we give the problem statement. We find an achievable region for broadcast problems over these networks and find the exact capacity region in the multiuser setup. Using this result we will give an outer bound on the capacity region of broadcast problems over wireless erasure networks which is tighter than the outer bounds that we get from the multicast type of arguments.
3.2
Problem Statement
The communication networks considered in this chapter, namely Wireless Erasure (WE) networks, are modeled by an acyclic directed graphG= (V,E) with node set V
and link (edge) set E ⊂ V × V. Each link (i, j)∈ E corresponds to a communication channel between the node i and node j. Looking back at Chapter 2, the input of all channels originating from node i is denoted by Xi chosen from input alphabet
X. The output of the communication channel corresponding to link (i, j) is denoted by Yij; Yij lies in alphabet set Yij. We assume interference-free property for all the incoming links to a node and denote the collection of received signal at node j by
Yj = (Yij, (i, j) ∈ E) ∈ Yj =4
Q
i:(i,j)∈EYij. Link (i, j) corresponds to an erasure
channel with probability of erasure ²ij in WE networks.
Let D = {d1, . . . , d|D|} denote the set of destination nodes and s be the source
node for the broadcast problem. Next, we define the class of block codes considered in this chapter.
A (d2nR1e, . . . ,d2nR|D|e, n) code for the broadcast problem in a wireless erasure network described in previous sections consists of the following components:
• A set of integers W(di) = {1,2, . . . ,d2nRie} that represent the message indices corresponding to information source intended for destination node di ∈ D. We assume that all the messages are equally likely. All the information sources are available at the source node indexed by s∈ V.
• An encoding function for the source nodes: fs :
Q
d∈DW(d) → Xsn.
• A set of encoding functions {fi,t}nt=1 for each node i 6= s ∈ V, where xi,t =
fi,t(yit−1) is the signal transmitted by node i at time t. Note that xi,t is a function of all the received symbols from all its incoming channels up to time
t−1.
• A decoding functiongdi at destination nodedi ∈ D,gdi :Ydin×{0,1}n|E| → W(di) such that
ˆ
w(di) =gdi(ydin,(sij,t,(i, j)∈ E,1≤t≤n)), (3.1) where ˆw(di) is the estimate of the message sent from sources based on received
signals at di and also the erasure occurrences on all the links of the network in the current block.
Note that Xi, Yij and Yi, all depend on the message vector w= (w(di), di ∈ D) that is being transmitted. Therefore, we will write them as Xi(w),Yij(w), and Yi(w) to specify what specific set of messages is transmitted.
We define the probability of error as the probability that the decoded message at one of the destinations is not equal to the transmitted message, i.e.,
Perr= Pr (∃di ∈ D: ˆW(di)6=W(di)). (3.2)
The set of rates (Ri, 1≤i ≤ |D|) is said to be achievable if there exist a sequence of (d2nR1e, . . . ,d2nR|D|e, n) codes such thatP
err→0 asn→ ∞. The capacity region,
C(G,{²ij, (i, j)∈ E}), is the set closure of the set of achievable rates.
In the remainder of this chapter we will look at the capacity region for broadcast problem in WE networks.