Analytical Method to Convert Circular Harmonic Expansion Coefficients for Sound Field Synthesis by Using Multipole Loudspeaker Array

We propose a method to synthesize the two-dimensional (2D) exterior sound field of a directional sound source using a Cartesian multipole loudspeaker array in which each loudspeaker unit is located on a Cartesian grid. We also propose an analytical method that models the sound field of the desired directional source in order to obtain weighting coefficients for each multipole from the circular harmonic expansion coefficients. The conversion method is derived by comparing the sound field created by a higher-order derivative of the free field Green's function and the corresponding field expressed by circular harmonic expansion coefficients in 2D space. In contrast to an existing analytical conversion method, the proposed method reproduces not only directivity patterns but also phases of the radiated sound from a target sound source, thereby enabling accurate sound field synthesis. We used numerical simulations to show that the proposed method achieved more accurate sound field reproduction than an existing pressure-matching-based method at higher frequency regions.