Integral transforms through the prism of distribution theory
Abstract: The Fourier transform is probably the most widely applied signal processing tool in science and engineering. It reveals the frequency composition of a time series by transforming it from the time domain into the frequency domain. However, it does not reveal how the signals frequency contents vary with time. A straightforward solution to overcoming the limitations of the Fourier transform the concept of the short-time Fourier transform (STFT). The short-time Fourier transform is a very effective device in the study of function spaces. However, significant barrier in application of the STFT is the fact that the fixed window function has to be predefined, which leads to a poor time-frequency resolution and, in general, the absence of a sufficiently good reconstruction algorithm. The Wavelet transform (WT) is used to overcome some of the shortcomings of the STFT. With the dilatation and translation of the window function, the WT has better phase modulation in the spectral domain. However, the self-similarity caused by the translation and the overlap in the frequency domain becomes non-avoidable since they do not permit straightforwardly the transfer of scale information into proper frequency information. The Stockwell transform (ST) also decomposes a signal into temporal and frequency components. In contrast to the WT, the ST exhibits a frequency-invariant amplitude response and covers the whole temporal axis creating full resolutions for each designated frequency. It is invertible, and recovers the exact phase and the frequency information without reconstructing the signal. The problem with the ST is its redundancy. But, there have been different strategies in order to improve the performance and the application of the ST.
On the other hand, the STFT, as a tool of the time-frequency analysis, contains localized time and frequency information of a function. Another idea is to localize information in time, frequency, and direction, which leads to directionally sensitive variant of STFT, which gives the Directional short time Fourier transform (DSTFT).
In mathematics, distributions extend the notion of functions. Distribution theory is a power tool in applied mathematics and the extension of integral transforms to generalized function spaces is an important subject with a long tradition. The theory is developed by proving that these transforms are well defined on the appropriate spaces of distribution. These is done by proving continuity results for these transforms on so called test function spaces, and then extending the definitions on distributions. In this talk, i consider several integrals transforms (STFT, WT, ST, DSTFT) and try to make short survey on their behaviour on distributions.
There are several approaches to the theory of distributions, but in all of them one quickly learn that distributions do not have point values, as functions do, despite the fact that they are called generalized functions. Natural generalization of this notion is the quasiasymptotic behavior of distributions. It is an old subject that has found applications in various fields of pure and applied mathematics, physics, and engineering. In the second part of my talk, I use Abelian and Tauberian ideas for asymptotic analysis of the mentioned integral transforms to characterize the asymptotic properties of a distribution.