Three Dimensional Sound Field Reproduction using Multiple Circular Loudspeaker Arrays: Functional Analysis Guided Approach

Abstract—Three dimensional sound field reproduction based on higher order Ambisonics requires the placement of loudspeakers on a sphere that surrounds the target reproduction region. The de-ployment of a spherical array is not trivial especially for implemen-tation in real rooms where the placement flexibility is highly desir-able. This paper proposes a design ofmultiple circular loudspeaker arrays for reproducing three dimensional sound fields originating from a limited region of interest. We apply a functional analysis framework to formulate the sound field reproduction problem in a closed form. Secondary source distributions and target sound fields are modeled as two Hilbert spaces and mapped by an integral op-erator and its adjoint operator, from which a self-adjoint operator is constructed and the singular value decomposition is applied to represent source distributions and sound fields with two sets of in-terrelated singular functions.We derive the solutions for a circular secondary source arrangement and propose the design of placing multiple circular loudspeaker arrays only over the limited region of interest. Such a design allows for non-spherical and non-uni-form loudspeaker placement and thus provides a flexible array ar-rangement. The reproduction accuracy of the proposed method is verified through numerical simulations. Index Terms—3D audio, ambisonics, circular loudspeaker ar-rays, sound field reproduction, spherical harmonics. I