# 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?

## 1 Answer

The linear canonical transforms are all area preserving, orientation preserving, linear transforms on the time-frequency plane. The fractional Fourier transforms are a subset, namely the rotations. Both sets form (Lie-)groups under composition, and the rotations are a subgroup.

• Can you please tell me what are the further subgroups apart from rotation? – Userhanu May 3 '16 at 10:59
• 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. – Jazzmaniac May 3 '16 at 11:33
• Thanks! Just one last request can you please tell me a book/reference where I can learn this SL(2,R) stuff, with a bit of background. – Userhanu May 3 '16 at 11:54
• Basic group theory is usually taught in linear algebra introduction classes. Beyond that point, groups appear in a lot of different contexts in mathematics, so it's difficult to point to a single resource. If you have the appropriate mathematical background, you should pick up a text book on group theory. It would probably be of advantage if it followed a geometric approach instead of the purely abstract algebraic one. I think the standard book on the fractional FT by Ozaktas et al. covers a bit of it too. – Jazzmaniac May 3 '16 at 12:46