  In this paper we address the problem of constructing a feature extractor
which combines Mallat's scattering transform framework with time-frequency
(Gabor) representations. To do this, we introduce a class of frames, called
uniform covering frames, which includes a variety of semi-discrete Gabor
systems. Incorporating a uniform covering frame with a neural network structure
yields the Fourier scattering transform $\mathcal{S}_\mathcal{F}$ and the
truncated Fourier scattering transform. We prove that $\mathcal{S}_\mathcal{F}$
propagates energy along frequency decreasing paths and its energy decays
exponentially as a function of the depth. These quantitative estimates are
fundamental in showing that $\mathcal{S}_\mathcal{F}$ satisfies the typical
scattering transform properties, and in controlling the information loss due to
width and depth truncation. We introduce the fast Fourier scattering transform
algorithm, and illustrate the algorithm's performance. The time-frequency
covering techniques developed in this paper are flexible and give insight into
the analysis of scattering transforms.
