--The fractional fourier transform(FRFT) which is a generalisation of classical fourier transform, was introduced number of years ago in the mathematics literature but appears to have remained largely unknown to a signal processing community to which it may, however, be a potentially useful. The fractional fourier transform depends on a parameter ‘α’ and can be interpreted as a rotation by an a...

We give a new characterization of the Gelfand triple of function spaces in (S0, L, S′ 0) by means of a family of time-frequency localization operators. The localization operators are defined by the short-time Fourier transform and determine the local time-frequency behavior, whereas the global timefrequency distribution is characterized by a sequence space norm. We also show that the alternativ...

1. A general criterion for pointwise Fourier inversion 2. Pointwise Fourier inversion on R n (n = 3) 3. Fourier inversion on R 2 4. Fourier inversion on R n (general n) 5. Fourier inversion on spheres 6. Fourier inversion on complex projective space, and variants 7. Fourier inversion on hyperbolic space, and variants 8. Fourier inversion on strongly scattering manifolds 9. Hermite expansions an...

The chirplet transform is the generalization form of fast Fourier Transform, short-time Fourier transform, and wavelet transform. It has the most flexible time frequency window and successfully used in practices. However, the chirplet transform has not inherent inverse transform, and can not overcome the signal reconstructing problem. In this paper, we proposed the improved chirplet transform (...

Wavelet transformation is one of the most practical mathematical transformations in the field of image processing, especially image and signal processing. Depending on the nature of the multiresolution analysis, Wavelet transformation become more accessible and powerful tools. In this paper, we refer to the mathematical foundations of this transformation.   Introduction: The...

Diagnostic of machine using the timefrequency representation of vibration is becoming widely used by many companies and experts. In recent years, the preventive maintenance is no more desirable and in fact it has evolved into the prediction maintenance. The current methods of machine diagnosis can not provide detailed diagnostics and condition prediction. This paper proposes the application of ...

We find optimal conditions on m-linear Fourier multipliers to give rise to bounded operators from a product of Hardy spaces Hj , 0 < pj ≤ 1, to Lebesgue spaces Lp. The conditions we obtain are necessary and sufficient for boundedness and are expressed in terms of L2-based Sobolev spaces. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky [1] and in the bilin...

The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known “qualitative” uncertainty principles into more quantitative estimates. We then show how to transfer this result to the discrete version of the Short Time Fourier Transform.