Differential Volterra filter: A two-stage decoupling method for audible sounds generated by parametric array loudspeakers based on Westervelt equation

Parametric array loudspeakers (PALs) produce highly directional sounds due to the parametric process in air. Its application in creating personal audio zones requires simple modeling of the audible sound field near the PAL, which is crucial for subsequent designs to reduce inherent nonlinear distortion. However, current accurate methods for describing the sound field, reliant on numerical solutions to wave equations, are computationally intensive. To achieve both effectiveness and simplicity, this paper proposes a time-domain model for audible sounds generated in the Westervelt far field, called the differential Volterra filter (Diff-VF). It is obtained through two stages: first, a narrow-band (NB) approximation is introduced to decouple the virtual source energy density from interactions of ultrasonic beams when solving the Westervelt equation. This results in a NB Westervelt solution for time-domain inputs. Second, to further develop a generic response independent of inputs, a temporal-spatial discretization is used to simplify the NB Westervelt solution into the Diff-VF model with a one-dimensional kernel. Numerical simulations confirmed the effectiveness of the NB Westervelt solution when compared with an exact solution, whether on- and off-axis. Experimental results validated that the Diff-VF model achieved superior prediction performance over existing VF-based models, with insensitivity to inputs and lower complexity.