  Fourier matrices naturally appear in many applications and their stability is
closely tied to performance guarantees of algorithms. The starting point of
this article is a result that characterizes properties of an exponential system
on a union of cubes in $\mathbb{R}^d$ in terms of a general class of Fourier
matrices and their extreme singular values. This relationship is flexible in
the sense that it holds for any dimension $d$, for many types of exponential
systems (Riesz bases, Riesz sequences, or frames) and for Fourier matrices with
an arbitrary number of rows and columns. From there, we prove new stability
results for Fourier matrices by exploiting this connection and using powerful
stability theorems for exponential systems. This paper provides a systematic
exploration of this connection and suggests some natural open questions.
