Integral formula for reproducing higher-order sound field with polar phase variation

Abstract

Recently, the reproduction of virtual sound sources with complex polar phase patterns has been attempted to provide various spatial attributes, such as the perceived width of a sound source. In this work, we deal with the problem of reproducing a higher-order circular harmonics of which phase alters along azimuth angle, and propose a line integral formula that can recreate the higher-order harmonic field on a horizontal plane with less reproduction errors. A line array of loudspeakers is considered, and a direct driving function for the line array is derived in terms of wave field synthesis (WFS). When the conventional WFS formula is applied to higher-order sound fields, however, significant reproduction errors are induced because target sound fields exhibits both rapid magnitude variations in elevation angle and abrupt phase variations in azimuthal angle, which can violate the underlying assumption of WFS. To circumvent this problem, a higher-order sound field with uniform polar phase variation at all elevation angles is considered as the target field, and a driving function for the line array is formulated. Simulations show that the presented driving function produces less reproduction errors than conventional formulas.