Field Reconstruction in Sensor Networks with Coverage Holes and Packet Losses

Environmental monitoring is often performed through wireless sensor networks, by randomly deploying sensor nodes over the geographical region of interest. Sensors sample a physical phenomenon (the so-called field) and send their measurements to a sink, which is in charge of reconstructing the field from such irregular samples. In this work, we focus on scenarios of practical interest where the sensor deployment is unfeasible in certain areas of the geographical region (e.g., due to terrain asperities), and the delivery of sensor measurements to the sink may fail (e.g., due to fading or to transmission collisions among sensors simultaneously accessing the wireless medium). Under these conditions, we carry out an asymptotic analysis and evaluate the quality of the estimation of a field defined over a d-dimensional (d>=1) domain when the sink uses linear filtering as a reconstruction technique. Specifically, given the matrix V representing the sampling system, we let the size of V go to infinity and its aspect ratio have a finite limit bounded away from zero. Then, we provide both the moments and density of the limiting spectral distribution of VV+, in terms of those obtained when the samples collected by the sink correspond to locations that are uniformly distributed over the geographical area. By using such asymptotic results, we approximate the mean square error on the estimated field through the eta-transform of VV+, and we derive the sensor network performance under the conditions described above.