Sampling Order-Limited Signals on the Sphere

We formulate and generalize a sampling scheme on the sphere based on the Gauss-Legendre (GL) quadrature that permits the accurate computation of the spherical harmonic transform (SHT) and its inverse for the signals band-limited at L and order-limited at M using L(2M + 1) number of samples. Signals that are nearly rotationally symmetric appear in many applications such as geophysics and antenna theory due to natural symmetry of the underlying process. For example, temperature variations in the climate data exhibit symmetry for a fixed co-latitude. Such signals when expanded in spherical harmonic domain do not have a strong dependence on spherical harmonics of higher orders. In this work, we propose a sampling method and develop associated SHT to save both the required number of samples and the computation cost while maintaining the signal reconstruction accuracy for order-limited signals. We also analyse numerical accuracy and computational complexity of the proposed transform and highlight the significance of proposed developments in the context of application in antenna theory.