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Continous Fractional Fourier Transform FrFT

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The FrFT, which is a generalization of the ordinary Fourier transform.



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The FRFT belongs to the class of time–frequency representations that have been extensively used by the signal processing community. In all the time–frequency representations, one normally uses a plane with two orthogonal axes corresponding to time and frequency. If we consider a signal x (t) to be represented along the time axis and its ordinary Fourier transform X(f) to be represented along the frequency axis, then the Fourier transform operator (denoted by F) can be visualized as a change in representation of the signal corresponding to a counterclockwise rotation of the axis by an angle π/2.
This is consistent with some of the observed properties of the Fourier transform. For example, two successive rotations of the signal through π /2 will result in an inversion of the time axis. Moreover, four successive rotations will leave the signal unaltered since a rotation through 2 π of the signal should leave the signal unaltered. The FRFT is a linear operator that corresponds to the rotation of the signal through an angle which is not a multiple of π /2, i.e. it is the representation of the signal along the axis ‘u’ making an angle a with the time axis. With the advent of FRFT and related concept, it is seen that the properties and applications of the ordinary Fourier transform are special cases of those of the FrFT.

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