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On the Short-Time Fourier Transform and Gabor Frames generated by B-splines
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
2012 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

In this thesis we study the short-time Fourier transform. The short-time Fourier transform of a function f(x) is obtained by restricting our function to a short time segment and take the Fourier transform of this restriction. This method gives information locally of f in both time and frequency simultaneously.To get a smooth frequency localization one wants to use a smooth window, whichmeans that the windows will overlap.

The continuous short-time Fourier transform is not appropriate for practical purpose, therefore we want a discrete representation of f. Using Gabor theory, we can write a function f as a linear combination of time- and frequency shifts of a fixed window function g with integer parameters a; b > 0. We show that if the window function g has compact support, then g generates a Gabor frame G(g; a; b). We also show that for such a g there exists a dual frame such that both G(g; a; b) and its dual frame has compact support and decay fast in the Fourier domain. Based on [2], we show that B-splines generates a pair of Gabor frames.

Place, publisher, year, edition, pages
2012. , 16 p.
Keyword [en]
short-time Fourier transform, time-frequency analysis, Gabor frames, B-splines
National Category
Mathematical Analysis
URN: urn:nbn:se:lnu:diva-20262OAI: diva2:535693
Subject / course
Physics, Chemistry, Mathematics
Available from: 2012-06-20 Created: 2012-06-20 Last updated: 2012-06-20Bibliographically approved

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