Fractional Fourier transform

The fractional Fourier transform (FRFT) is a linear transformation generalizing the continuous Fourier transform, and 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 (1980), but it was not widely recognized until it was independently reinvented in 1993 by several groups of researchers (Almeida, 1994).

A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber (1991) as another name for the z-transform, especially for the particular case that corresponds to a discrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linear chirp). See also the chirplet transform. (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.

Formal 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 ${\mathcal {F}}^{(n+1)}(f)={\mathcal {F}}({\mathcal {F}}^{n}(f))$. The FRFT further extends this definition to handle non-integer powers $n=2\alpha /\pi$ for any real $\alpha$, denoted by ${\mathcal {F}}_{\alpha }(f)$ and having the properties:

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

when $n=2\alpha /\pi$ 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 $\alpha =\pi /2$, this becomes precisely the continuous Fourier transform, and for $\alpha =-\pi /2$ it is the inverse continuous Fourier transform.

If $\alpha$ is an even or odd integer multiple of $\pi$, then the cotangent and cosecant functions above diverge. However, this can be handled by taking the limit.

There also exist related fractional generalizations of similar transforms such as the discrete Fourier transform.

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).
• 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).
• 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 z-transform, not the FRFT.)