# What is the relation between the domain of Fractional Fourier transform and time and frequency?

This refers to Wikipedia's page on Fractional Fourier Transform.
In the definition section it clearly writes that:

The FrFT argument $$u$$ is neither a spatial one $$x$$ nor a frequency $$\xi$$. We will see why it can be interpreted as linear combination of both coordinates $$(x,\xi)$$

But I did not see the relation between the domain of Fractional Fourier transform ($$u$$) and space and frequency $$(x,\xi)$$. I can understand that domain '$$u$$' is a linear combination of '$$(x,\xi)$$' but my question what is the relation? Can anyone point out the exact relation?

• up arrow on this question. Commented Oct 20, 2015 at 22:50

I think it is more correct to say that the $$u$$ axis is a linear combination of the $$x$$ and $$\xi$$ axes:

$$(x, \xi) = (1, 0)x + (0, 1)\xi = \left((1, 0)\cos(\alpha) + (0, 1)\sin(\alpha)\right)u,$$

that is:

$$x = u\cos(\alpha)\\ \xi = u\sin(\alpha).$$

The definition of the fractional Fourier transform from that Wikipedia page is:

$$\mathcal{F}_\alpha[f](u) = \sqrt{1-i\cot(\alpha)} e^{i \pi \cot(\alpha) u^2} \int_{-\infty}^\infty e^{-i2\pi (\csc(\alpha) u x - \frac{\cot(\alpha)}{2} x^2)} f(x)\, \mathrm{d}x.$$

If you use another variable, $$y$$, for the integral, and substitute $$u = \sqrt{x^2 + \xi^2}$$ and $$\alpha = \text{atan}(\xi, x)$$, then the fractional Fourier transform can be written as:

$$\mathcal{F}[f](x, \xi) = e^{i \pi \frac{\xi}{x} (x^2+\xi^2)} \sqrt{1-\frac{i \xi}{x}}\int^\infty_{-\infty}e^{i \pi \frac{y}{x} \left(\xi y-2 x^{\frac{1}{\xi}} \sqrt{\xi^2+x^2}\right)} f(y)\, \text{d}y.$$

• Can we gain any insight by change of variable? I knew that relation to x but did not realize the relation to frequency. Commented Oct 21, 2015 at 23:58
• Dunno, but it's the same as going from polar notation to rectangular notation. Commented Oct 22, 2015 at 0:07
• Olli, this question (and the Wikipedia definition) intrigue me. I don't quite get it. Commented Jun 12 at 5:30
• @robertbristow-johnson After 9 years I find the Wikipedia illustration upload.wikimedia.org/wikipedia/en/b/b3/… more accessible. That kind of 45 degrees time-frequency plane rotation I guess you could implement by complex exponential modulation (a shear of the time-frequency plane), Fourier transform, complex exponential modulation (another shear). Commented Jun 12 at 5:59
• Just as long as if the number of fractional FTs adds up to an integer multiple of 4, you get the original function back. Commented Jun 12 at 6:04