Energy Savings for Virtual MISO in Fractal Sensors Networks

We design a model of wireless terminals obtained from a Poisson point process with support in an embedded fractal map. The terminals, transmitters and receivers, form a virtual MISO (Multiple Input Single Output) system with successful reception under SNR (signal-tonoise ratio) capture condition. We show that if we omit antennas cross sections, the energy needed to broadcast a packet of information tends to zero when the density of transmitters and receivers increases. This property is a direct consequence of the fact that the support map is fractal and would not hold if the terminal distribution were Poisson uniform, as shown by simulations. Moreover, the result is rather surprising since one could expect that the energy to have a packet correctly captured by the all receivers must increase proportionally with the number of receivers yet this quantity is negligible if the antennas cross sections are small. As a consequence, the result stops to be valid if the cross sections overlap or if we consider a masking effect due to antennas, which would imply an extremely large density of terminals. In the case where the cross sections of the transmitters have a non-zero value, the energy has a non-zero limit which decays to zero when the cross sections tend to zero.