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eventually there will be a node so close to the source that it alone will be able to make the decision. All that is required then is the capability for the higher SNR nodes to suppress activity by nodes that detect the phenomenon at lower delity but at SNRs at levels where they might be inclined to engage in cooperative detection. A simple relay strategy will then suf ce to get the information out of the network. Notice that the keys to the scalability are: separation of function between source coding and long-range communication (allowing different densities for the logical functions of communications and detection), local decision-making, and a delity criterion. Without the last two, the amount of information to convey would outstrip communications capability for certain source models. Now clearly neither the communications relay strategy nor the local decision rules are optimal in information theoretic senses; they are merely suf cient to ensure scalability. However, this result is highly suggestive of what an optimal strategy might look like under a delity (quality of service) constraint. Once a suf cient number of nodes are identi ed that (with appropriate network source coding) achieve mutual information between observations and source phenomenon above some threshold, then no more nodes need be involved. Long-range transport of data need not further consider interactions of source and channel coding. If the network is to be scalable, local communications must dominate, with longer-range communications becoming much less important (e.g., not requiring heroic cooperation strategies). Cooperation in relaying the data may be more of a secondary matter of achieving reliable communications over local gaps in the network. Given the local interactions dominate, it is in this domain that combined source coding and routing/channel coding are of most interest. Put another way, while there is no general source-channel separation theorem, when source models, deployment scenarios, and QoS ( delity) objectives are speci ed, the above sensor network scalability results imply that there can be separation on particular scales. For example, if the objective is to identify and track a number of objects that enter a study region, and the signals decay strongly with distance, then only the nodes that are in close proximity to each object will have suf cient mutual information to be included in the decision-making. While some performance loss may result in separating source and channel coding to the point of data fusion among this group, beyond this the problem reduces to a pure network coding (or routing) problem, with no further consideration of source coding. Furthermore, in the limit of very strong decay of signals with distance, the relative SNRs of the best and next best sensor will in likelihood be quite large so that even optimal fusion buys little performance gain, allowing separation of source and channel coding even at the local level. This suggests a two-scale design approach: For local regions, a solution speci c to the underlying physical model is devised, which may involve some combined source coding/routing/channel coding scheme, while for long-range transport only communications considerations apply. The corresponding optimization problem can be conceived of as a two-step process. The rst step is the selection of a cost function that will control the set of nodes that will be locally cooperating, given a cooperation protocol. One may then select the cooperation protocol from a set of candidates. For a signi cant set of practical scenarios
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the number of nodes involved and their time of involvement are the most important resource questions. For example, in short-range radio communications, power is consumed simply from the fact of a radio being on, with transmission power being a secondary issue. The power and bandwidth requirements linearly depend upon their duty cycle. Similarly, the time spent processing is related to the amount of data being collected, again a function of the duty cycle of involvement. The amount of processing (or communications) required is often an exponential function of the number of nodes involved, again suggesting that minimization of the number of nodes involved (while meeting the delity constraint) is a primary consideration. This type of cost function also leads to a reasonable set of optimization problems, as follows. The scaling laws for the number of nodes to be involved in fusing information about some source event to the desired delity can be relatively easily obtained in many circumstances. In particular, this is true for both maximal ratio (MR) combining and beamforming problems. For example, if the delity requirement is that the fused sensor data must achieve an SNR greater than some threshold in detecting some point source, and if AWGN is assumed and there is a second power source propagation loss, then assuming that coherent combining is possible every time the radius of the cooperation region is doubled, the expected SNR improves 6 dB (i.e., it scales with area, assuming a uniform distribution of nodes) [1, 5]. The relationship between coverage and density may similarly be easily obtained for this set of assumptions. For other models and cooperation protocols, the details of the tradeoff will vary, but the above serves as a good prototypical scenario for assessing the suitability of particular cost functions and also conveniently serves as a lower bound on achievable performance. Assuming that there is some maximum cost in terms of the number of nodes involved in the fusion activity, one would obtain a family of curves of the type illustrated in Figure 14.3 for different delity requirements. Below density l we hit the ceiling nmax on the maximum number n of nodes permitted, resulting in an inability to meet the delity requirements, while above density h only the minimum number of nodes nmin to carry out the task are needed (e.g., one in detection, four for 3-D localization, etc.). The cost function must include the resource cost for the network as a whole in having a higher density (that is, the total number of nodes in the network, N), which re ects the equipment and maintenance costs. This will drive the solution
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