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I have seen this plot on Wikipedia's Fractional Fourier Transform, where it discusses the rotations between frequency and time domains of a distribution, however I do not understand how to plot a distribution on its time and frequency 3d plot or even what the distribution would look like with time and frequency as its input variables. How does one plot the frequency and time domain of a distribution on a 3D plot?

Time-Frequency Plot

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This may be better addressed as a computational question and I'm just guessing here, but let's see what happens if I just get started...

To make to make the plots above, you need to map each bin in one space (say the time domain) to a intensity function in the other space (frequency in our example). This is accomplished with a limited application of the (plain 'ole) Fourier transform. The full transform is $$ \hat{f}(\xi) = \int_{-\infty}^{\infty} \mathrm{d}x \,f(x) \exp \left( -2\pi i x \xi \right) \,, $$ which involves the whole of the domain space, but we are interested in knowing the range space intensity due to a limited interval $[a,b)$ in the domain. So we simply limit the integral to that bin $$ \hat{f}_{a,b}(\xi) = \int_{a}^{b} \mathrm{d}x \,f(x) \exp \left( -2\pi i x \xi \right) \,, $$ or equivalently convolve the input function with a band pass function $$ \hat{f}_{a,b}(\xi) = \int_{-\infty}^{\infty} \mathrm{d}x \,f(x) \exp \left( -2\pi i x \xi \right) \left[ \Theta(x-a)\Theta(x+b)\right]\,, $$ (where $\Theta$ is the Heaviside step function).

If you make your bins too small this the convolution begins to resemble a delta function with undesirable results: $$ \begin{align*} \hat{f}_{a,b}(\xi) &= \int_{-\infty}^{\infty} \mathrm{d}x \,f(x) \exp \left( -2\pi i x \xi \right) \delta(x-a)\\ &= f(a) \exp \left( -2\pi i a \xi \right) \,, \end{align*} $$ because you end with a simple sinusoidal result.

There may be a computational advantage to using a less abrupt filter than the (simplest) one that I have exhibited here. Indeed the blurry edges of the above figure suggest that this may be the case.

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  • $\begingroup$ I was curious, because I was thinking of this signal diagram in terms of Quantum Mechanics wave-functions. Does it make sense to plot the momentum space and position space as orthogonal axes? $\endgroup$ – linuxfreebird Sep 14 '14 at 16:30
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    $\begingroup$ Hmmm. "Make sense" depends on what you want to do with the results. I can't think of a physics application for this right off, and am really beginning to suspect that the question should be moved to either Computational Science or Mathematics. But maybe one of our theorists knows better. $\endgroup$ – dmckee --- ex-moderator kitten Sep 14 '14 at 16:32

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