On Bandlimited Signal Reconstruction From the Distribution of Unknown Sampling Locations

In remote sensing with a large array of spatially distributed sensors, localization of individual sensor nodes could be difficult. Motivated by the smart-dust paradigm, this work addresses the reconstruction of spatially bandlimited fields from samples taken at unknown but statistically distributed sampling locations. Periodic one-dimensional bandlimited fields are considered for sampling. Samples of the field at uniform independent and identically distributed locations are obtained. The statistical realization of the sampling locations is not known. First, it is shown that a bandlimited field cannot be uniquely determined with samples taken at uniformly distributed but unknown locations, even if the number of samples is infinite. Next, it is assumed that the order of sample locations is known. With order information on sample locations, consistent bandlimited field estimates are designed and analyzed for mean-squared error. The proposed estimates are analyzed for mean-squared error for two cases-when the field measurements are noiseless and when the field measurements are affected by additive independent noise process with finite variance. Finally, a central-limit is established for the field estimate in the case where field measurements are noiseless.