# Multidimensional Sampling of Isotropically Bandlimited Signals

## TL;DR

This paper derives a new lower bound on the reconstruction error for multidimensional isotropically bandlimited signals sampled on arbitrary lattices, linking optimal sampling lattices to sphere covering and packing properties.

## Contribution

It introduces a precise lower bound for error variance in multidimensional sampling and establishes a rigorous connection between optimal sampling lattices and sphere covering/packing.

## Key findings

- Lower bound is exact at high and low sampling rates.
- Optimal lattice at low rates is dual of the best sphere-covering lattice.
- Optimal lattice at high rates is dual of the best sphere-packing lattice.

## 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.