On Bandlimited Field Estimation from Samples Recorded by a Location-Unaware Mobile Sensor

Sampling of physical fields with mobile sensor is an emerging area. In this context, this work introduces and addresses some aspects of a fundamental question: can a spatial field be estimated from samples taken at unknown sampling locations? In a field (signal) sampling setup, unknown sampling locations, sample quantization, unknown bandwidth of the field, and presence of measurement-noise present difficulties in the process of field estimation. In this work, except for quantization, the other three issues will be tackled together in a mobile-sampling framework. Spatially bandlimited fields are considered. It is assumed that measurement-noise affected field samples are collected on spatial locations obtained from an unknown renewal process. That is, the samples are obtained on locations obtained from a renewal process, but the sampling locations and the renewal process distribution are unknown. In this challenging unknown sampling location setup, it is shown that the mean-squared error in field estimation decreases as O(1/n) where n is the average number of samples collected by the mobile sensor. The average number of samples collected is determined by the inter-sample spacing distribution in the renewal process. An algorithm to ascertain spatial field's bandwidth is detailed, which works with high probability as the average number of samples n increases. This algorithm works in the same setup, i.e., in the presence of measurement-noise and unknown sampling locations.