Everything about Fractional Fourier Transform totally explained
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 (1980), but it wasn't widely recognized until it was independently reinvented around
1993 by several groups of researchers (Almeida, 1994).
A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber (1991) 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 (for example 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
is denoted by
must be simply
or
for
an
even or odd multiple of
, respectively.
There also exist related fractional generalizations of similar transforms such as the
discrete Fourier transform.
Physical Meaning of the Fractional Fourier Transform
The physical meaning of the Fourier transform is as a transformation of a time domain signal into a frequency domain signal. On the other hand, the physical meaning of the inverse Fourier transform is as a transformation of a frequency domain signal into a time domain signal. Apparently, fractional Fourier transform can transform a signal (either in the time domain or frequency domain) into the domain between time and frequency.
Take the below figure as an example. If the signal in the time domain is rectangular (as below), it'll 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.
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.
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 doesn't overlap with the desired signal in the time frequency domain. Let’s see the following example. We can't 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.
Further Information
Get more info on 'Fractional Fourier Transform'.
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