# How is Linear Canonical Transform a generalization of Fractional Fourier Transform?

I have studied that Fourier transform changes the domain of a signal from time to frequency, and in that way it is a 90 degree shift. When it comes to Fractional Fourier Transform a generalization of Fourier Transform, the resulting transform can lie anywhere between time and frequency domain depending on parameter 'alpha'.

My question is how in the same way Linear Canonical Transform is a generalization of Fractional Fourier Transform.Its physical interpretation in terms of signal processing. Can any body help?

• As a transformation on time-frequency points, the linear canonical transforms are $\mathrm{SL}(2,\mathbf{R})$. There are infinitely many subgroups, but the most important in terms of signal processing are the cyclic subgroup of order 4 generated by the Fourier transform, the $\mathrm{SO}(2)$ of the fractional Fourier transform and the $\mathrm{GL}(1)$ of time/frequency scaling. Commented May 3, 2016 at 11:33