The Fractional Fourier Transform and Its Application to Energy Localization Problems

• Main Authors: ter Morsche Hennie G, Oonincx Patrick J
• Format: Article
• Language: English
• Published: SpringerOpen 2003-01-01
• Series: EURASIP Journal on Advances in Signal Processing
• Subjects: fractional Fourier transform; Wigner distribution; symplectic transformation; energy localization
• Online Access: http://dx.doi.org/10.1155/S1110865703305086

<p/> <p>Applying the fractional Fourier transform (FRFT) and the Wigner distribution on a signal in a cascade fashion is equivalent to a rotation of the time and frequency parameters of the Wigner distribution. We presented in ter Morsche and Oonincx, 2002, an integral representation formula that yields affine transformations on the spatial and frequency parameters of the <inline-formula><graphic file="1687-6180-2003-246759-i1.gif"/></inline-formula>-dimensional Wigner distribution if it is applied on a signal with the Wigner distribution as for the FRFT. In this paper, we show how this representation formula can be used to solve certain energy localization problems in phase space. Examples of such problems are given by means of some classical results. Although the results on localization problems are classical, the application of generalized Fourier transform enlarges the class of problems that can be solved with traditional techniques.</p>