Title: Multidimensional Sampling of Isotropically Bandlimited Signals,
Abstract: A new lower bound on the average reconstruction error variance of
multidimensional sampling and reconstruction is presented. It applies to
sampling on arbitrary lattices in arbitrary dimensions, assuming a stochastic
process with constant, isotropically bandlimited spectrum and reconstruction by
the best linear interpolator. The lower bound is exact for any lattice at
sufficiently high and low sampling rates. The two threshold rates where the
error variance deviates from the lower bound gives two optimality criteria for
sampling lattices. It is proved that at low rates, near the first threshold,
the optimal lattice is the dual of the best sphere-covering lattice, which for
the first time establishes a rigorous relation between optimal sampling and
optimal sphere covering. A previously known result is confirmed at high rates,
near the second threshold, namely, that the optimal lattice is the dual of the
best sphere-packing lattice. Numerical results quantify the performance of
various lattices for sampling and support the theoretical optimality criteria.