Loudspeaker 3D directivity estimation with first order microphone measurements on a 2D plane

This paper proposes an efficient method to estimate the 3D directivity pattern of loudspeakers or portable devices with embedded speakers. We place the loudspeaker on a horizontal turntable and use a first order microphone located on the horizontal plane to measure pressure and pressure gradients along 3 orthogonal directions to construct equivalent virtual arrays of first order microphones on the horizontal plane. By exploiting the properties of the associated Legendre functions, we construct the 3D directivity pattern of the loudspeaker over frequencies. The method is equivalent to having a measurement setup consist of a dense spherical array encompassing the loudspeaker. The underlying theory and method are corroborated by simulations as well as measurements of the directivity of a physical loudspeaker.