Higher-Order Autocorrelations on Finite Abelian Groups

TL;DR

This paper establishes new bounds on the amount of higher-order autocorrelation data needed to identify signals on finite abelian groups, with implications for signal reconstruction in various fields.

Contribution

It introduces two novel upper bounds on data requirements for signal determination on finite abelian groups, considering Fourier transform properties.

Findings

Derived upper bounds depending on Fourier transform vanishing

Classified signals on Z_6 not determined by fifth-order data

Provided examples on Z_30 illustrating lower bounds

Abstract

The question of determining a signal from its higher-order autocorrelation data is of practical interest in fields as varied as X-ray crystallography, image processing, and satellite communications. At the heart of the issue is how much of this autocorrelation data one truly needs. We prove two new upper bounds on the order of data needed to determine a signal on a general (i.e. not necessarily cyclic) finite abelian group depending on some knowledge of the vanishing of the signal's Fourier transform. In investigating lower bounds on the required data, we classify signals on $\mathbb{Z}_6$ not determined by their fifth-order data and provide analogous examples on $\mathbb{Z}_{30}$.