Plane wave radial filter design using pre-emphasized time-domain representations

Abstract

The spatial encoding, manipulation and decoding of a sound field in the spherical harmonics domain requires a discrete-time realization of the radial dependent parts, called radial filters. For plane waves, the spectra of the radial basis functions are described by the spherical Bessel functions and the impulse responses by the Legendre polynomials. Although the radial filters can be designed efficiently by sampling the time-domain radial functions, the resulting spectrum typically suffers from aliasing. In this paper, we present a radial filter design method where the aliasing is reduced by means of spectral pre-emphasis. The antiderivatives of the radial functions are derived in closed form by exploiting the Rodrigues’ formula. This leads to a spectral shaping of the radial functions that is inversely proportional to the frequency. Since the energy lying beyond the Nyquist limit is attenuated, the pre-emphasized signal can be sampled with reduced aliasing. The original magnitude spectrum is then restored by applying a differentiation to the sampled signal. In this study, a first-order IIR filter is used as a digital differentiator. The aliasing reduction achieved by the proposed method is demonstrated for different radii and antiderivative orders.

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ORCID for Nara Hahn: orcid.org/0000-0003-3564-5864

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