Fractional Fourier transform

In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. F α [ f ] ( u ) = 1 − i cot ⁡ ( α ) e i π cot ⁡ ( α ) u 2 ∫ − ∞ ∞ e − i 2 π ( csc ⁡ ( α ) u x − cot ⁡ ( α ) 2 x 2 ) f ( x ) d x {displaystyle {mathcal {F}}_{alpha }(u)={sqrt {1-icot(alpha )}}e^{ipi cot(alpha )u^{2}}int _{-infty }^{infty }e^{-i2pi left(csc(alpha )ux-{frac {cot(alpha )}{2}}x^{2} ight)}f(x),mathrm {d} x} In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the n-th power, where n need not be an integer — thus, it can transform a function to any intermediate domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since then, there has been a surge of interest in extending Shannon's sampling theorem for signals which are band-limited in the Fractional Fourier domain. A completely different meaning for 'fractional Fourier transform' was introduced by Bailey and Swartztrauber as essentially another name for a z-transform, and in particular for the case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear chirp) and evaluating at a fractional set of frequency points (e.g. considering only a small portion of the spectrum). (Such transforms can be evaluated efficiently by Bluestein's FFT algorithm.) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT. The continuous Fourier transform F {displaystyle {mathcal {F}}} of a function ƒ: R → C is a unitary operator of L2 that maps the function ƒ to its frequential version ƒ̂ (all expressions are taken in the L2 sense, rather than pointwise): and ƒ is determined by ƒ̂ via the inverse transform F − 1 {displaystyle {mathcal {F}}^{-1}} Let us study its n-th iterated F n {displaystyle {mathcal {F}}^{n}} defined by F n [ f ] = F [ F n − 1 [ f ] ] {displaystyle {mathcal {F}}^{n}={mathcal {F}}]} and F − n = ( F − 1 ) n {displaystyle {mathcal {F}}^{-n}=({mathcal {F}}^{-1})^{n}} when n is a non-negative integer, and F 0 [ f ] = f {displaystyle {mathcal {F}}^{0}=f} . Their sequence is finite since F {displaystyle {mathcal {F}}} is a 4-periodic automorphism: for every function ƒ, F 4 [ f ] = f {displaystyle {mathcal {F}}^{4}=f} . More precisely, let us introduce the parity operator P {displaystyle {mathcal {P}}} that inverts x {displaystyle x} , P [ f ] : x ↦ f ( − x ) {displaystyle {mathcal {P}}colon xmapsto f(-x)} . Then the following properties hold: The FrFT provides a family of linear transforms that further extends this definition to handle non-integer powers n = 2α/π of the FT.