Causal Spatio-Temporal Sound Field Reconstruction

In sound field control applications, it is commonly assumed that one has access to an accurate representation of the sound field in the region of interest. This is a problematic assumption since the reconstruction of a sound field from available microphone measurements is especially challenging in real-time applications where only causal measurements are available. Notably, causal time-windowed observations introduce correlation between frequency components, making sound field reconstruction methods that process each frequency band independently sub-optimal. In this work, we formulate a causal finite-window spatio-temporal linear minimum mean-square error estimator for sound field reconstruction. The sound field is modeled as the solution to the wave equation driven by a stationary stochastic spatio-temporal source distribution, which induces a physically interpretable covariance function. It is shown that this covariance function is closely related to the classical diffuse-field coherence model. Since the computational complexity grows rapidly with the number of spatio-temporal observations, we formulate a budget-constrained spatio-temporal sample selection approach to minimize the posterior reconstruction variance. The proposed estimator and sampling strategy are evaluated using both simulated and measured sound fields, demonstrating improved short-window reconstruction compared to frequency domain finite-window baselines.