A Compressed Sensing Framework for Dynamic Sound-Field Measurements

The conventional sampling of sound fields by use of stationary microphones is impractical for large bandwidths. Satisfying the Nyquist-Shannon sampling theorem in three-dimensional space requires a huge number of sampling positions. Dynamic sound-field measurements with moving microphones together with a compressed-sensing recovery allow for weakening the spatial sampling problem. For bandlimited signals, the dynamic samples taken along the microphone trajectory may be related to the room impulse responses on a virtual grid in space via spatial interpolation. The tracking of the microphone positions and the knowledge of the excitation sequence allow for setting up a linear system of equations that can be solved for the room impulse responses on the modeled virtual grid. Nevertheless, there is still the necessity for recovering a huge number of sound-field variables, in order to ensure aliasing-free reconstruction. Thus, for practical applications, random or suboptimally chosen trajectories may be expected to lead to underdetermined sampling problems for a given volume of interest. In this paper, we present a compressed sensing framework that enables us to uniquely solve the dynamic sampling problem despite having underdetermined variables. The spatio-temporal sampling problem is integrated into compressed sensing models that allow for stable and robust sub-Nyquist sampling given incoherent measurements. For a modeled equidistant grid and sparse Fourier representations, the influence of the microphone trajectories on the compressed sensing problem is investigated and a simple expression is derived for evaluating trajectories with regard to compressed-sensing based recovery.