Fractional Fourier transform
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,[1] by solving for the Green's function for phase-space rotations, and also by Namias,[2] generalizing work of Wiener[3] on Hermite polynomials.
However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups.
[4] Since then, there has been a surge of interest in extending Shannon's sampling theorem[5][6] 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[7] 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 FRFT provides a family of linear transforms that further extends this definition to handle non-integer powers
For any real α, the α-angle fractional Fourier transform of a function ƒ is denoted by
For α = π/2, this becomes precisely the definition of the continuous Fourier transform, and for α = −π/2 it is the definition of the inverse continuous Fourier transform.
We will see why it can be interpreted as linear combination of both coordinates (x,ξ).
The α-th order fractional Fourier transform operator,
Define the scaling and chirp multiplication operators as follows:
turns out to be a scaled and chirp modulated version of
Here again the special cases are consistent with the limit behavior when α approaches a multiple of π.
The Fourier transform is essentially bosonic; it works because it is consistent with the superposition principle and related interference patterns.
[17] Because quantum circuits are based on unitary operations, they are useful for computing integral transforms as the latter are unitary operators on a function space.
On the other hand, the interpretation of the inverse Fourier transform is as a transformation of a frequency domain signal into a time domain signal.
The fractional Fourier transform is a rotation operation on a time–frequency distribution.
The following figure shows the results of the fractional Fourier transform with different values of α.
The fractional Fourier transform is equivalent to the Fresnel diffraction equation[19] [20].
Fractional Fourier transform can be used in time frequency analysis and DSP.
[21] It is useful to filter noise, but with the condition that it does not overlap with the desired signal in the time–frequency domain.
We cannot apply a filter directly to eliminate the noise, but with the help of the fractional Fourier transform, we can rotate the signal (including the desired signal and noise) first.
We then apply a specific filter, which will allow only the desired signal to pass.
In contrast, using time domain or frequency domain tools without a fractional Fourier transform would only allow cutting out rectangles parallel to the axes.
Fractional Fourier transforms also have applications in quantum physics.
For example, they are used to formulate entropic uncertainty relations,[22] in high-dimensional quantum key distribution schemes with single photons,[23] and in observing spatial entanglement of photon pairs.
[24] They are also useful in the design of optical systems and for optimizing holographic storage efficiency.
