  We derive a compressive sampling method for acoustic field reconstruction
using field measurements on a predefined spherical grid that has theoretically
guaranteed relations between signal sparsity, measurement number, and
reconstruction accuracy. This method can be used to reconstruct band-limited
spherical harmonic or Wigner $D$-function series (spherical harmonic series are
a special case) with sparse coefficients. Contrasting typical compressive
sampling methods for Wigner $D$-function series that use arbitrary random
measurements, the new method samples randomly on an equiangular grid, a
practical and commonly used sampling pattern. Using its periodic extension, we
transform the reconstruction of a Wigner $D$-function series into a
multi-dimensional Fourier domain reconstruction problem. We establish that this
transformation has a bounded effect on sparsity level and provide numerical
studies of this effect. We also compare the reconstruction performance of the
new approach to classical Nyquist sampling and existing compressive sampling
methods. In our tests, the new compressive sampling approach performs
comparably to other guaranteed compressive sampling approaches and needs a
fraction of the measurements dictated by the Nyquist sampling theorem.
Moreover, using one-third of the measurements or less, the new compressive
sampling method can provide over 20 dB better denoising capability than
oversampling with classical Fourier theory.
