Detecting quasicrystals with quadratic time-frequency distributions

The usefulness of time-frequency analysis methods in the study of quasicrystals was pointed out in [3], 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 extend significantly those of [3], covering 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 particular, the qualitative improvement of the second case with respect to [3] is that the support of the matrix-Wigner transform of $\mu$ is only supposed to lie in a product of discrete sets, obtaining uniform discreteness of support and spectrum as a consequence.