Short-time Fourier transform laser Doppler holography

We report a demonstration of laser Doppler holography at a sustained acquisition rate of 250 Hz on a 1 Megapixel complementary metal-oxide-semiconductor (CMOS) sensor array and image display at 10 Hz frame rate. The holograms are optically acquired in off-axis configuration, with a frequency-shifted reference beam. Wide-field imaging of optical fluctuations in a 250 Hz frequency band is achieved by turning time-domain samplings to the dual domain via short-time temporal Fourier transformation. The measurement band can be positioned freely within the low radio-frequency spectrum by tuning the frequency of the reference beam in real-time. Video-rate image rendering is achieved by streamline image processing with commodity computer graphics hardware. This experimental scheme is validated by a non-contact vibrometry experiment

Super-Resolution from Short-Time Fourier Transform Measurements

While spike trains are obviously not band-limited, the theory of super-resolution tells us that perfect recovery of unknown spike locations and weights from low-pass Fourier transform measurements is possible provided that the minimum spacing, $\Delta$, between spikes is not too small. Specifically, for a cutoff frequency of $f_c$, Donoho [2] shows that exact recovery is possible if $\Delta > 1/f_c$, but does not specify a corresponding recovery method. On the other hand, Cand\`es and Fernandez-Granda [3] provide a recovery method based on convex optimization, which provably succeeds as long as $\Delta > 2/f_c$. In practical applications one often has access to windowed Fourier transform measurements, i.e., short-time Fourier transform (STFT) measurements, only. In this paper, we develop a theory of super-resolution from STFT measurements, and we propose a method that provably succeeds in recovering spike trains from STFT measurements provided that $\Delta > 1/f_c$.Comment: IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), May 2014, to appea

Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation property and having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.Comment: 14 page

A variational restriction theorem

We establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short survey of the recent field of maximal Fourier restriction theory.Comment: 10 pages, v2: bibliography is updated, a short survey of the maximal Fourier restriction is include

The short-time Fourier transform of distributions of exponential type and Tauberian theorems for shift-asymptotics

We study the short-time Fourier transform on the space $\mathcal{K}_{1}'(\mathbb{R}^n)$ of distributions of exponential type. We give characterizations of $\mathcal{K}_{1}'(\mathbb{R}^n)$ and some of its subspaces in terms of modulation spaces. We also obtain various Tauberian theorems for the short-time Fourier transform.Comment: 17 page

Zeros of the Wigner Distribution and the Short-Time Fourier Transform

We study the question under which conditions the zero set of a (cross-) Wigner distribution W (f, g) or a short-time Fourier transform is empty. This is the case when both f and g are generalized Gaussians, but we will construct less obvious examples consisting of exponential functions and their convolutions. The results require elements from the theory of totally positive functions, Bessel functions, and Hurwitz polynomials. The question of zero-free Wigner distributions is also related to Hudson's theorem for the positivity of the Wigner distribution and to Hardy's uncertainty principle. We then construct a class of step functions S so that the Wigner distribution W (f, 1 (0,1)) always possesses a zero f $\in$ S $\cap$ L p for p < $\infty$, but may be zero-free for f $\in$ S $\cap$ L $\infty$. The examples show that the question of zeros of the Wigner distribution may be quite subtle and relate to several branches of analysis