Analysis of the sphericalwave truncation error for spherical harmonic soundfield expansions

Three dimensional soundfield recording and reproduction is an area of ongoing investigation and its implementation is increasingly achieved through use of the infinite Spherical Harmonic soundfield expansion. Perfect recording or reconstruction requires infinite microphones or loudspeakers, respectively. Thus, real-world approximations to both require spatial discretisation, which truncates the soundfield expansion and loses some of the soundfield information. The resulting truncation error is the focus of this paper, specifically for soundfields comprising of spherical waves. We define two norms of the truncation error to signal ratio, L₂ and L_∞, for comparison and use in different situations. Finally we observe how some of these errors converge to the plane wave case under certain circumstances.