Detecting quasicrystals with quadratic time-frequency distributions

TL;DR

This paper advances the detection of quasicrystal structures using quadratic time-frequency distributions, especially matrix-Wigner transforms, extending previous work on Wigner transforms and including the Ambiguity function.

Contribution

It introduces new methods using matrix-Wigner transforms to identify quasicrystal structures, covering a broad class of quadratic time-frequency distributions.

Findings

Classical Wigner transform and Cohen class are effective in detecting quasicrystals.

Support and spectrum of measures are shown to be uniformly discrete under certain conditions.

The Ambiguity function can be used to detect quasicrystals even when classical methods do not.

Abstract

The usefulness of time-frequency analysis methods in the study of quasicrystals was pointed out in a previous paper, where we proved that a tempered distribution $\mu$ on ${\mathbb R}^d$ whose Wigner transform is a measure supported on the cartesian product of two uniformly discrete sets in ${\mathbb R}^d$ is a Fourier quasicrystal. In this paper we go further in this direction using the matrix-Wigner transforms to detect quasicrystal structures. The results presented here cover essentially all the most important quadratic time-frequency distributions, and are obtained considering two different (disjoint) classes of matrix-Wigner transforms, discussed respectively in Theorems 1 and 2. The transforms considered in Theorem 1 include the classical Wigner transform, as well as all the time-frequency representations of matrix-Wigner type belonging to the Cohen class. On the other hand Theorem 2, which does not apply to the classical Wigner, has, as main example, the Ambiguity function. In this second case we only suppose that the support of the matrix-Wigner transform of $\mu$ is contained in the cartesian product of two discrete sets, obtaining that both the support and the spectrum of $\mu$ are uniformly discrete.