Limitations in the extrapolation of wave fields from circular measurements

This paper analyzes the constraints in extrapolating sound fields from measurements using circular microphone setups. Different representations of 2D wave fields are compared and several derivations for the decomposition into circular harmonics are summarized. Theoretical limits are derived for the order of the obtainable circular harmonic components due to spatial sampling, the measuring radius, and sensor noise. The consequences of order limitations are addressed for the extrapolation of wave fields to different radii. It is shown that even though an exact extrapolation is only possible within the measuring aperture, a coarse extrapolation can be possible beyond the radius of the microphone setup depending on the direction relative to the wave propagation. For this purpose far field approximations are derived for the extrapolation of an order limited decomposition of a plane wave in different directions. Real world measurements show good agreement with these investigations.