A survey of uncertainty principles and some signal processing applications

The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, emphasize their mutual connections and investigate practical consequences. The discussion is strongly oriented towards, and motivated by signal processing problems, from which significant advances have been made recently. Relations with sparse approximation and coding problems are emphasized

Two Aspects of the Donoho-Stark Uncertainty Principle

We present some forms of uncertainty principle which involve in a new way localization operators, the concept of $\varepsilon$-concentration and the standard deviation of $L^2$ functions. We show how our results improve the classical Donoho-Stark estimate in two different aspects: a better general lower bound and a lower bound in dependence on the signal itself.Comment: 20 page

A planar large sieve and sparsity of time-frequency representations

With the aim of measuring the sparsity of a real signal, Donoho and Logan introduced the concept of maximum Nyquist density, and used it to extend Bombieri's principle of the large sieve to bandlimited functions. This led to several recovery algorithms based on the minimization of the $L_{1}$-norm. In this paper we introduce the concept of {\ planar maximum} Nyquist density, which measures the sparsity of the time-frequency distribution of a function. We obtain a planar large sieve principle which applies to time-frequency representations with a gaussian window, or equivalently, to Fock spaces, $\mathcal{F}_{1}\left( \mathbb{C}\right)$, allowing for perfect recovery of the short-Fourier transform (STFT) of functions in the modulation space $M_{1}$ (also known as Feichtinger's algebra $S_{0}$) corrupted by sparse noise and for approximation of missing STFT data in $M_{1}$, by $L_{1}$-minimization

Designing Gabor windows using convex optimization

Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal l2-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time-frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler-Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found