A Modal Analysis of Spatial Discretization of Spherical Loudspeaker Distributions Used for Sound Field Synthesis

The theory of sound field synthesis methods like Wave Field Synthesis (WFS) and Near-field Compensated Higher Order Ambisonics (NFC-HOA) may be formulated based on the assumption of continuous distributions of secondary sources that enclose the receiver area in the general case. In practice, a finite number of discrete loudspeakers is used, which constitutes a fundamental departure from the theoretical requirements. In this paper, we present a detailed analysis of the consequences of this spatial discretization on the synthesized sound field via an analytical frequency-dependent modal decomposition of the latter for the case of Gaussian sampling. It is shown that the underlying mechanisms are essentially similar to those in the discretization of circular secondary source distributions so that the results obtained in the latter context may be directly applied. The outstanding parallel in the discretization of spherical and circular distributions is the fact that in both cases, repetitions in a given space frequency domain occur. Therefore, the spatial bandwidth of the desired sound field has essential influence on the properties of the evolving artifacts. We propose to categorize sound field synthesis methods into spatially narrowband, wideband, and fullband approaches and show that NFC-HOA constitutes a spatially narrowband method and that WFS constitutes a spatially fullband method.