Time Domain Sampling of the Radial Functions in Spherical Harmonics Expansions

Spherical harmonics representations are widely adopted in applications such as analysis, manipulation, and synthesis of wave fields that are either captured or simulated. Recent studies have shown that, for broadband fields, a time domain representation of spherical harmonics expansions can benefit from computational efficiency and favorable transient properties. For practical usage, an accurate discrete-time modeling of spherical harmonics expansions is indispensable. The main challenge is to model the so called radial functions which describe the radial and frequency dependencies. In homogeneous cases, they are commonly realized as finite impulse response filters where the coefficients are obtained by sampling the continuous-time representations. This article investigates the temporal and spectral properties of the resulting discrete-time radial functions. The spectral distortions caused by aliasing are evaluated both analytically and numerically, revealing the influence of the distance from the expansion center, sampling frequency, and fractional sample delay. It is also demonstrated how the aliasing can be reduced by employing a recently introduced band limitation method.