Abstract
We present structural results on optimal number of clusters and optimal rate with respect to long haul distance in a cluster based cooperative multiple input and multiple output (MIMO) sensor network. Using the concepts of super and sub modular functions and their associated properties it has been proved that the number of clusters to minimize the energy consumption is a decreasing function of long haul distance. From this, we conclude that as the long haul distance increases the cooperation among the nodes increases to counter the path loss due to fading in the channel. Furthermore, we also prove the rate to minimize the energy consumption is a decreasing function of long haul distance. Finally, through our numerical results we observe that optimizing node cooperation can lead to large scale energy savings.
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