fractional Fourier transform ：< FRET >

In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a linear transformation 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 an 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 given by Namias[1], but it was not widely recognized until it was independently reinvented around 1993 by several groups of researchers[2].
A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber[3] 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.
See also the chirplet transform for a related generalization of the Fourier transform.

Contents[hide]
1 Definition
1.1 Related transforms
2 Interpretation of the Fractional Fourier Transform
3 Application
4 See also
5 External links
6 References
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[edit] Definition
If the continuous Fourier transform of a function f(t) is denoted by , then , and in general ; similarly, denotes the n-th power of the inverse transform of F(ω). The FRFT further extends this definition to handle non-integer powers n = 2α / π for any real α, denoted by and having the properties:

when n = 2α / π is an integer, and:

More specifically, is given by the equation:

Note that, 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.
If α is an integer multiple of π, then the cotangent and cosecant functions above diverge. However, this can be handled by taking the limit, and leads to a Dirac delta function in the integrand. More easily, since , must be simply f(t) or f( − t) for α an even or odd multiple of π, respectively.

[edit] Related transforms
There also exist related fractional generalizations of similar transforms such as the discrete Fourier transform. The discrete fractional Fourier transform is defined in (Candan, Kutay & Ozaktas 2000) and (Ozaktas, Zalevsky & Kutay 2001, Chapter 6).

[edit] Interpretation of the Fractional Fourier Transform
Further information: Linear canonical transformation
The usual interpretation of the Fourier transform is as a transformation of a time domain signal into a frequency domain signal. 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. Apparently, fractional Fourier transforms can transform a signal (either in the time domain or frequency domain) into the domain between time and frequency: it is a rotation in the time-frequency domain. This perspective is generalized by the linear canonical transformation, which generalizes the fractional Fourier transform and allows linear transforms of the time-frequency domain other than rotation.
Take the below figure as an example. If the signal in the time domain is rectangular (as below), it will become a sinc function in the frequency domain. But if we apply the fractional Fourier transform to the rectangular signal, the transformation output will be in the domain between time and frequency.

Fractional Fourier Transform
Actually, fractional Fourier transform is a rotation operation on the time frequency distribution. From the definition above, for α=0, there will be no change after applying fractional Fourier transform, and for α=π/2, fractional Fourier transform becomes a Fourier transform, which rotates the time frequency distribution with π/2. For other value of α, fractional Fourier transform rotates the time frequency distribution according to α. The following figure shows the results of the fractional Fourier transform with different values of α.

Time/Frequency Distribution of Fractional Fourier Transform.

[edit] Application
Fractional Fourier transform can be used in time frequency analysis and DSP. It is useful to filter noise, but with the condition that it does not overlap with the desired signal in the time frequency domain. Let’s see the following example. 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. Thus the noise will be removed completely. Then we use the fractional Fourier transform again to rotate the signal back and we can get the desired signal.

Fractional Fourier Transform in DSP.
Thus, using just truncation in the time domain, or equivalently low-pass filters in the frequency domain, one can cut out any convex set in time-frequency space; just using time domain or frequency domain methods without fractional Fourier transforms only allow cutting out rectangles parallel to the axes.

[edit] See also
Other time-frequency transforms:
Linear canonical transformation
short-time Fourier transform
wavelet transform
chirplet transform

[edit] External links
DiscreteTFDs -- software for computing the fractional Fourier transform and time-frequency distributions
"Fractional Fourier Transform" by Enrique Zeleny, The Wolfram Demonstrations Project.
Dr YangQuan Chen's FRFT (Fractional Fourier Transform) Webpages

[edit] References
^ V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," J. Inst. Appl. Math. 25, 241–265 (1980).
^ Luís B. Almeida, "The fractional Fourier transform and time-frequency representations," IEEE Trans. Sig. Processing 42 (11), 3084–3091 (1994).
^ D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Review 33, 389-404 (1991). (Note that this article refers to the chirp-z transform variant, not the FRFT.)
Ozaktas, Haldun M.; Zalevsky, Zeev; Kutay, M. Alper (2001), The Fractional Fourier Transform with Applications in Optics and Signal Processing, Series in Pure and Applied Optics, John Wiley & Sons, http://www.ee.bilkent.edu.tr/~haldun/wileybook.html
Candan, C.; Kutay, M.A.; Ozaktas, H.M. (May 2000), "The discrete fractional Fourier transform", IEEE Transactions on Signal Processing 48 (5): 1329–1337, doi:10.1109/78.839980
A. W. Lohmann, "Image rotation, Wigner rotation and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181–2186 (1993).
Soo-Chang Pei and Jian-Jiun Ding, "Relations between fractional operations and time-frequency distributions, and their applications," IEEE Trans. Sig. Processing 49 (8), 1638–1655 (2001).
*Jian-Jiun Ding, Time frequency analysis and wavelet transform class notes, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
Retrieved from "http://en.wikipedia.org/wiki/Fractional_Fourier_transform"


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Fractional Fourier Transform

There are two sorts of transforms known as the fractional Fourier transform.

The linear fractional Fourier transform is a discrete Fourier transform in which the exponent is modified by the addition of a factor ,
However, such transforms may not be consistent with their inverses unless is an integer relatively prime to so that . Fractional fourier transforms are implemented in Mathematica as Fourier[list, FourierParameters -> a, b], where is an additional scaling parameter. For example, the plots above show 2-dimensional fractional Fourier transforms of the function for parameter ranging from 1 to 6.
The quadratic fractional Fourier transform is defined in signal processing and optics. Here, the fractional powers of the ordinary Fourier transform operation correspond to rotation by angles in the time-frequency or space-frequency plane (phase space). So-called fractional Fourier domains correspond to oblique axes in the time-frequency plane, and thus the fractional Fourier transform (sometimes abbreviated FRT) is directly related to the Radon transforms of the Wigner distribution and the ambiguity function. Of particular interest from a signal processing perspective is the concept of filtering in fractional Fourier domains. Physically, the transform is intimately related to Fresnel diffraction in wave and beam propagation and to the quantum-mechanical harmonic oscillator.
SEE ALSO: Ambiguity Function, Discrete Fourier Transform, Fourier Transform, Phase Space, Radon Transform, Time-Space Frequency Analysis, Wigner Distribution
Portions of this entry contributed by Haldun M. Ozaktas s