Fourier sliding-window

A robust detection system should be able to provide both frequency and spatial information. While frequency serves to identify the presence of a defect, spatial information is necessary to identify the location of defect. It is well known that the Fourier transform suffers from a lack of spatial information while Gabor filtering methods are very computational expensive since the 2D convolution between the image and filter must be carried out in a sliding window throughout the entire image. These limitations could be eliminated by wavelets. 

Spectrograms. To obtain deeper insight into the dynamics of the wave packet, it is of great interest to know the temporal evolution of the frequency components, i.e. the time Ai when and with what intensity I a certain frequency u) occurs in the real-time spectra. This information can easily be extracted from spectrograms I At,uj) [51, 262, 327]. For this purpose a procedure is used which first was proposed by Stolow et al. to calculate spectrograms of transient data [51]. A sliding-window Fourier transform... 

The windowed Fourier transform [2,3] (also called the short time Fourier transform) was introduced so that the frequency information about a signal could be localised with respect to time. Instead of analysing the function f(t) as a whole, the windowed Fourier transform performs a Fourier transform on pieces of the function. The pieces are obtained by using a windowing function G(t) which slides across the function. The windowed Fourier transform of f(t) is defined as... 

The windowed Fourier coefficients are functions of frequency o), as well as the translated position of the window b. Fig. 3 demonstrates how the window slides across the function, and thus, how it is possible to obtain information about the frequency content of the signal over some constant time interval. 

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