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Wireless sensors have several constraints, such as a short transmission range, poor processing capabilities, and a limited available energy. Our goal is to design an energy efficient sensor network by exploiting the concept of collaborative signal processing. We consider a network composed of multiple sensor nodes, each of which corresponds to a processor. By using an appropriate collaborative computational algorithm and communication scheme, we make sensors operate as a Distributed Digital Signal Processor (DDSP) and generate the desired results. The benefit of such an approach is twofold: (i) the energy and computational limitations of the individual sensors can be overcome, and (ii) for a given level of processing complexity, the energy efficiency of the overall network can increase significantly. We apply the DDSP approach to the Fast Fourier Transform algorithm, and derive results showing the improvement in performance that can be obtained through the proposed technique. Moreover, we study the existing trade-off between energy saving, data latency and number of employed sensors.