Sampling smooth spatio-temporal physical fields: When will the aliasing error increase with time?

Abstract

Acquisition of physical fields, such as temperature along a path, using a distributed array of sensors (samples) is of interest. For smooth spatial fields, in a Nyquist style sampling setup, the aliasing error is determined by the (spatial) spectral profile of a field. Physical fields and their spectral properties evolve with time. In this work, the spectral evolution of spatio-temporal fields is analyzed, where the field is given by physical law comprising of a constant coefficient linear partial differential equation. A procedure to examine whether field's spatial spectral profile will become worse with time, from aliasing point of view, is developed. The procedure is exemplified using a second-order PDE in this work. The analysis is extended to include simple point source terms. Techniques such as the Fourier transform, the unilateral Laplace transform, and root-locus plot from control theory are utilized in this work.