Windowed linear canonical transform: its relation to windowed Fourier transform and uncertainty principles

The windowed linear canonical transform is a natural extension of the classical windowed Fourier transform using the linear canonical transform. In the current work, we first remind the reader about the relation between the windowed linear canonical transform and windowed Fourier transform. It is shown that useful relation enables us to provide different proofs of some properties of the windowed linear canonical transform, such as the orthogonality relation, inversion theorem, and complex conjugation. Lastly, we demonstrate some new results concerning several generalizations of the uncertainty principles associated with this transformation.

Research of vibrations of vertical component of cutting force during turning

A method for localizing bands of vertical component of the cutting force in time spectrum during metal turning has been established. The method for determining the source of spectral components is based on the windowed Fourier transform. The powers of the spectral components of vibration signals at different cutting speeds in especially sensitive frequency ranges are compared.

Multifrequency beam-based migration in inhomogeneous media using windowed Fourier transform frames

SUMMARY We consider the problem of inhomogeneous subsurface imaging using beam waves. The formulation is based on the ultra-wide-band phase-space beam summation (UWB-PS-BS) method, which is structured upon windowed Fourier transform (WFT) expansions of surface fields and sources. In this approach, the radiated field is given as a superposition of beam propagators. Here, we use the beams first for expanding the surface sources and the scattered data, and then for imaging where we use the backpropagation and cross-correlation of beams. This formulation enables a target oriented imaging approach, where we take into account only pairs of source and receiver beams that pass near a region of interest, and thus extract only the relevant data arriving from this region. It also leads to a priori sparse representation of both the beam domain data and the beam propagators. A physical cogent for the beam domain data is obtained under the Born approximation. The beam domain data can be approximated as the local interaction between the beam propagators and the medium reflectivity. Thus, one may interpret the beam domain data as a local Snell’s law reflection in the direction defined by the vector summation of the incident beam and backpropagated beam ray parameters. We demonstrate a physical model for the beam domain data and the salient features of the proposed imaging algorithm using numerical examples.

Partial discharge feature extraction based on synchrosqueezed windowed Fourier transform and multi-scale dispersion entropy

The recognition of partial discharge mode is an important indicator of the insulation condition in transformers, based on which maintenance can be arranged. Discharge feature extraction is the key to recognize discharge mode. To solve the problem of poor stability and low recognition rate of partial discharge mode, this paper proposes a feature extraction method based on synchrosqueezed windowed Fourier transform and multi-scale dispersion entropy. First, the four partial discharge signals collected under laboratory conditions are decomposed by synchrosqueezed windowed Fourier transform, then a number of band-limited intrinsic mode type functions are obtained, and the original feature quantities of partial discharge signals are obtained by calculating the multi-scale dispersion entropies of each intrinsic mode type function. Based on that, original feature quantity is optimized by using the maximum relevance and minimum redundancy criteria. Finally, the classification is implemented by the support vector machine. Experimental results show that in the case of noise interference, the proposed synchrosqueezed windowed Fourier transform–multi-scale dispersion entropy method can still accurately describe the feature of different discharge signals and has a higher recognition rate than both the empirical mode decomposition–multi-scale dispersion entropy method and the direct multi-scale dispersion entropy method.

Uncertainty principles related to quaternionic windowed Fourier transform

In this paper, we first introduce uncertainty principles for the quaternion Fourier transform (QFT). We then provide a different proof of the well-known properties of the quaternionic windowed Fourier transform (QWFT) using properties of the QFT which is a little bit simpler than usual. Based on uncertainty principles for the QFT and the relationship between the QFT and QWFT, we establish uncertainty principles related to the QWFT.