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The fractional Fourier transform is a rotation operation on a time–frequency distribution. From the definition above, for α = 0, there will be no change after applying the fractional Fourier transform, while for α = π/2, the fractional Fourier transform becomes a plain Fourier transform, which rotates the time–frequency distribution with ...
The fractional Fourier transform corresponds to rotation by an arbitrary angle; they are the elliptic elements of SL 2 (R), represented by the matrices [] = [ ]. The Fourier transform is the fractional Fourier transform when θ = 90 ∘ . {\displaystyle \theta =90^{\circ }.}
The theory of fractional integration for periodic functions (therefore including the "boundary condition" of repeating after a period) is given by the Weyl integral. It is defined on Fourier series, and requires the constant Fourier coefficient to vanish (thus, it applies to functions on the unit circle whose integrals evaluate to zero). The ...
The Fourier transform is suitable to filter out the noise that is a combination of sinusoid functions. If signal are not separable in both time and frequency domains, using the fractional Fourier transform (FRFTs) is suitable to filter out the noise that is a combination of higher order exponential functions.
The pseudocode below performs the GS algorithm to obtain a phase distribution for the plane "Source", such that its Fourier transform would have the amplitude distribution of the plane "Target". The Gerchberg-Saxton algorithm is one of the most prevalent methods used to create computer-generated holograms. [2]
Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used in time–frequency analysis. [28] In physics, this transform was introduced by Edward Condon. [29]
A signal, as a function of time, may be considered as a representation with perfect time resolution.In contrast, the magnitude of the Fourier transform (FT) of the signal may be considered as a representation with perfect spectral resolution but with no time information because the magnitude of the FT conveys frequency content but it fails to convey when, in time, different events occur in the ...
The essence of transform theory is that by a suitable choice of basis for a vector space a problem may be simplified—or diagonalized as in spectral theory. Main examples of transforms that are both well known and widely applicable include integral transforms [ 1 ] such as the Fourier transform , the fractional Fourier Transform , [ 2 ] the ...