Bandlimited Spatiotemporal Field Sampling with Location and Time Unaware Mobile Sensors

Abstract

Sampling of physical fields has been a topic that been studied extensively in literature but has been restricted to a small class of fields like temperature or pollution which are essentially modelled by the standard second order partial differential equation for diffusion. Furthermore, a large number of sampling techniques have been studied from sensor networks to mobile sampling under a variety of conditions like known and unknown locations of the sensors or sampling locations and with samples affected by measurement and/or quantization noise. Also, certain works have also addressed time varying fields incorporating the difference in known timestamps of the obtained signals.It would be of great interest to explore fields which are modelled by more general constant coefficient linear partial differential equations to address a larger class of fields that have a more complex evolution. Additionally, this works address an extremely general and challenging problem, in which such a field is sampled using an inexpensive mobile sensor such that both, the locations of the samples and the timestamps of the samples are unknown. Moreover, the locations and timestamps of the samples are assumed to be realizations of two independent unknown renewal processes. Furthermore, the samples have been corrupted by measurement noise. In such a challenging setup, the mean squared error between the original and the estimated signal is shown to decreasing as O(1/n), where n is the average sampling density of the mobile sensor.