We extend an uncertainty principle due to Beurling into a characterization of Hermite functions. More precisely, all functions f on R which may be written as P (x) exp(Ax, x), with A a real symmetric definite positive matrix, are characterized by integrability conditions on the product f(x)f̂(y). We also give the best constant in uncertainty principles of Gelf’and Shilov type. We then obtain sim...

In low voltage distribution networks, there is a high risk of fires caused by wiring faults that lead to short circuits. Arcing protection devices (ARCs) already exist, but experiments show they also often do not disable such faults. The operation most existing arc fault detection based on the frequency current components and their comparison with load profiles. For fast recognition low-frequen...

We propose an algorithm that uses the discrete wavelet transform (DWT) as a tool to compute the discrete Fourier transform (DFT). The Cooley-Tukey FFT is shown to be a special case of the proposed algorithm when the wavelets in use are trivial. If no intermediate coeecients are dropped and no approximations are made, the proposed algorithm computes the exact result, and its computational comple...

Based on the windowed Fourier transform, the windowed Fourier ridges (WFR) algorithm and the windowed Fourier filtering algorithm (WFF) have been developed and proven effective for fringe pattern analysis. The WFR algorithm is able to estimate local frequency and phase by assuming the phase distribution in a local area to be a quadratic polynomial. In this paper, a general and detailed statisti...

In this article, we introduce the notion of two-sided quaternion wave-packet transform which inherits advantages both windowed Fourier and wavelet transforms with some additional promising features. The preliminary analysis encompasses derivation fundamental properties including, orthogonality relation, energy preserving inversion formula range theorem by utilizing machinery transforms. Besides...

This invited paper – of a tutorial and review character – presents an overview of two classes of time-frequency signal representations. The first class, in which the signal arises linearly, deals with the windowed Fourier transform and its sampled version (also known as the Gabor transform) and the inverse of the latter: Gabor’s signal expansion. We will show how Gabor’s signal expansion and th...