Fractional Fourier transform

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, but it was not widely recognized until it was independently reinvented around 1993 by several groups of researchers. Since then, there has been a surge of interest in extending Shannon's sampling theorem for signals which are bandlimited in 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.

See also the chirplet transform for a related generalization of the Fourier transform.

Definition

If the continuous Fourier transform of a function f(t) is denoted by $\mathcal{F}(f)$, then $\mathcal{F}^2(f)=\mathcal{F}(\mathcal{F}(f))$, and in general $\mathcal{F}^{(n+1)}(f)=\mathcal{F}(\mathcal{F}^n(f))$; similarly, $\mathcal{F}^{-n}(F)$ denotes the n-th power of the inverse transform $\mathcal{F}^{-1}(F)$ of F(ω). The FRFT further extends this definition to handle non-integer powers n = 2α / π for any real α, denoted by $\mathcal{F}_\alpha(f)$ and having the properties:

$\mathcal{F}_\alpha(f) = \mathcal{F}^{2\alpha/\pi}(f)$

when n = 2α / π is an integer, and:

$\mathcal{F}_{\alpha+\beta}(f) = \mathcal{F}_\alpha(\mathcal{F}_\beta(f)) = \mathcal{F}_\beta(\mathcal{F}_\alpha(f)).$

More specifically, $\mathcal{F}_\alpha(f)$ is given by the equation:

$\mathcal{F}_\alpha(f)(\omega) = \sqrt{\frac{1-i\cot(\alpha)}{2\pi}} e^{i \cot(\alpha) \omega^2/2} \int_{-\infty}^\infty e^{-i\csc(\alpha) \omega t + i \cot(\alpha) t^2/2} f(t)\, dt.$

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 $\mathcal{F}^2(f)=f(-t)$, $\mathcal{F}_\alpha(f)$ must be simply f(t) or f( − t) for α an even or odd multiple of π, respectively.

Related transforms

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

Fractional wavelet transform: A generalization of the conventional wavelet transform in the fractional Fourier transform domains.

Generalization

The Fourier transform is essentially bosonic; it works because it is consistent with the superposition principle and related interference patterns. There is also a fermionic Fourier transform. These have been generalized into a supersymmetric FRFT, and a supersymmetric Radon transform. There is also a fractional Radon transform, a symplectic FRFT, and a symplectic wavelet transform. 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. A quantum circuit has been designed which implements the FRFT.

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.

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. Consider 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.

See also

Other time-frequency transforms:

• Linear canonical transformation
• short-time Fourier transform
• wavelet transform
• chirplet transform

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

Bibliography

• 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, ISBN 0471963461, 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.

• Saxena, R., Singh, K., (2005) Fractional Fourier transform: A novel tool for signal processing, J. Indian Inst. Sci., Jan.–Feb. 2005, 85, 11–26. http://journal.library.iisc.ernet.in/vol200501/paper2/11.pdf.