![]() ![]() ![]() In signal processing, time–frequency analysis comprises those techniques that study a signal in both the time and frequency domains simultaneously, using various time–frequency representations. More recently, many authors have been interested in the DSTFT. Building on this idea, in Giv defined the directional short-time Fourier transform (DSTFT) and proved several orthogonality results and reconstruction formulas. It's well known that the Radon transform and its inverse problem have a wide range of applications, namely to partial differential equations, group representations, X-ray technology, nuclear magnetic, resonance scanning and also in many other physical and astrophysical fields, for more details about the Radon transform we refer the reader to. In Candes and Donoho introduced a version of wavelet called ridgelets in order to study a frequency spectrum of a given signal as well in a certain time interval, as in a certain direction, later Grafakos and Sansing extend this notion by using the directionally sensitive Radon transform defined as the average of a compactly supported function over hyperplanes with respect to the natural hypersurface measure. In the sixties the time-frequency analysis has emerged with the works of Gabor who provided an interesting way to study the local frequency spectrum of signals by introducing many time-frequency representations, as for instance, the short-time Fourier transform, the Wigner Ville distribution or also the ambiguity radar function where all of theses representations have a same commun point, that is the simultaneous representation of the space and the frequency variables in a same set called the time-frequency plane, for more details about time frequency analysis, we refer the reader to. In the last decades, the general idea of the uncertainty has been interpreted differently by many authors who have given many formulations of the localization and the smallness, we cite for instance, Faris-Price local uncertainty principle which formulation could be viewed in somehow as a generalization of the Heisenberg inequality, Beckner uncertainty principle who give a lower bound of the sum of the entropies of a given function and its Fourier transform or also Donoho-Stark uncertainty principle studying a form of concentration related to a function and its Fourier transform, moreover uncertainty principles have many important applications in signal analysis, for more details, we refer the reader to. In harmonic analysis, the uncertainty principles state that a nonzero function f and its Fourier transform f cannot be at the same time simultaneously and sharply localized, that is, it's impossible for a nonzero function and its Fourier transform to be arbitrary small at the same time. The idea of uncertainty was first introduced by Heisenberg few decades ago in quantum mechanics, it states that we can't localize simultaneously with an arbitrary precision the position and the momentum of a high speed particle by providing a lower bound of the product of their variances. ![]()
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