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Figure 4.5. Waiting for the last blocks to ll.
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different receivers. However, it has a drawback in the receipt overhead. In fact, since transmission loops through different blocks, after having completely received and decoded a certain number of blocks, a receiver can potentially receive and discard packets belonging to the blocks already decoded before receiving the missing packets necessary to decode the last blocks. The receipt overhead is measured in terms of unnecessary received packets (see Figure 4.5). Starting from the achievements of references 8, 22 and 23, an implementation of RS codes has been provided also for wireless sensor networks. Kim et al. [9] propose a point-to-point data transmission protocol for WSNs that combines multiple strategies, namely, (i) erasure codes, (ii) retransmission of lost packets, and (iii) a route- x technique. RS codes are used to add redundancy to transmission such that the receiver is able to reconstruct original data despite losses or congestion (up to a certain point). Nevertheless, as underlined in reference 9, retransmissions cannot be eliminated completely. Rather, it is necessary to provide integration between encoding and retransmission of data and to ne-tune this integration for better ef ciency. Retransmissions are managed hop-by-hop such that the point of retransmission is moved progressively forward toward the destination node. Experimental results show that these two strategies together are very ef cient and result in good reliability. Nevertheless, it has also been noted that both link-level retransmissions and erasure codes have poor performance in case of route failures. When a route is no longer available, consecutive losses occur and retransmissions are of no use because the route toward the destination does not exist any more. Neither erasure codes are helpful in this case because receivers cannot receive the minimum amount of encoded messages that is needed to reconstruct the original messages. A viable solution consists in nding an alternative route as soon as possible. Kim et al. [9] have also integrated route xing into the framework with encoding and retransmission of data and have used the Beacon Vector Routing (BVR) protocol [33] (a particular case of geographic routing for WSNs) at the routing layer to provide the special support needed. Results show improvements in both reliability of transmissions and RTTs experienced by messages. 4.3.2 Tornado Codes Tornado codes were introduced in 1997 to be fast, loss resilient codes, suitable to errorprone transmission channels. A preliminary version of these codes was presented in
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reference 26, and further improvements were later added in reference 27. The overall structure of Tornado codes is related to the low-density parity-check codes introduced by Gallagar [34]. Tornado codes cannot overtake the information-recovery capacity of RS codes, which is already optimal, but focus on faster encoding and decoding times. Namely, given the length of the output block of the encoder corresponding to an input block of length k (k < n), encoding and decoding times of Tornado codes are of order O(n ln(1/ )) for some > 0 (instead of O(n2 ) as in RS-codes). The price paid for much faster encoding and decoding times, with respect to RS codes, is that arrival of k packets is no longer suf cient for the receiver to reconstruct the source data. Encoding Process. Suppose the source data to be encoded is a vector of k symbols in the nite eld GF(q) and let a symbol be a bit. The encoding process to yield the correspondent Tornado code C relies on a bipartite graph B. Let the overall code be referred to as C(B). It consists of a systematic code and thus includes the k bits of the original source message, named message bits, as well as k redundant bits (0 < < 1) named check bits. The bipartite graph consists of k left nodes, k right nodes, and some edges to connect left nodes to right nodes. Left nodes correspond to message bits and right nodes correspond to check bits. Check bits are obtained by XOR-ing over the source message bits according to the bipartite graph B. Namely, a check bit is worked out by XOR-ing over all the message bits it is connected to in the graph B (see Figure 4.6a). The bipartite graph is sparse, random, and irregular. Its construction is actually a bit complex, and it is principally oriented to the reduction of decoding costs (see below). By making things a bit more complex than has been described so far, a Tornado code is generally obtained with a multi-bipartite graph that is, a sequence (cascade) of successive bipartite graphs. In the rst stage, a set of k redundant bits
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Figure 4.6. (a) A bipartite graph de nes a mapping from message bits to check bits. (b) Bits x1 , x2 , and c are used to solve for x3 .
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