Higher-Order Anti-Derivatives of Band Limited Step Functions for the Design of Radial Filters in Spherical Harmonics Expansions

This paper presents a discrete-time model of the spherical harmonics expansion describing a sound field. The so-called radial functions are realized as digital filters, which characterize the spatial impulse responses of the individual harmonic orders. The filter coefficients are derived from the analytical expressions of the time-domain radial functions, which have a finite extent in time. Due to the varying degrees of discontinuities occurring at their edges, a time-domain sampling of the radial functions gives rise to aliasing. In order to reduce the aliasing distortion, the discontinuities are replaced with the higher-order anti-derivatives of a band-limited step function. The improved spectral accuracy is demonstrated by numerical evaluation. The proposed discrete-time sound field model is applicable in broadband applications such as spatial sound reproduction and active noise control.