Fractional Fourier Transform - Interpretation of The Fractional Fourier Transform

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.

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.