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I'm confused about a tutorial problem concerning the Fourier transform of the $\textrm{sinc}^2$ function.

Specifically, the question involves the Fourier transform of $\textrm{sinc}^2(100\pi t)$, where I would have assumed the transform property $f(at) \iff \frac{1}{|a|}F(\frac{\omega}{a})$ would apply. Similarly, every transform table I can find shows that $\textrm{sinc}^2(at)\iff \frac{1}{a}\triangle(\frac{\omega}{a})$

But apparently $\textrm{sinc}^2(100\pi t)\iff \frac{1}{100}\triangle(\frac{\omega}{400\pi})$

I would greatly appreciate if anyone could explain to me where I'm going wrong with the rules of the Fourier transform and how it works in this instance.

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    $\begingroup$ Hints: the Fourier transform of a sinc is a rect in the frequency domain. Squaring a time-domain sinc corresponds to convolving the rect in the frequency domain with itself which results in a $\Delta$ the length of whose support is twice the length of the support of the rect. $\endgroup$ Nov 23, 2021 at 4:51
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    $\begingroup$ It's hard (actually impossible) to answer this question without knowing which definition of the sinc function you use (there are at least two common ones), and without knowing how you define the triangle function. $\endgroup$
    – Matt L.
    Nov 23, 2021 at 11:10

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The problem with this question, and with comparing different Fourier transform tables, is that you have to make sure you know which definitions apply. If you look up the wikipedia page on the sinc function, you'll see that there are two common definitions: $$\textrm{sinc}(x)=\frac{\sin(x)}{x}\tag{1}$$ and $$\textrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x}\tag{2}$$

In DSP, we usually use definition $(2)$, but if you look up transform pairs, you have to make sure you understand which definition is being used.

Similarly, there is no single accepted definition of the triangular function. It could be a triangle with a base of width $1$ or $2$. Are you sure you know which definition was used by the authors of the different tables you're referring to?

Moreover, which definition of the Fourier transform are you referring to? Is it the same in all tables that you are comparing? Is it unitary or non-unitary, ordinary frequency or angular frequency?

IF you use definition $(2)$ of the sinc function, if you define the triangular function $\textrm{tri}(x)$ as a symmetric triangle of height $1$ with a base width of $2$, and if you use the unitary form of the Fourier transform with ordinary frequency, then I can assure you that the following relation holds:

$$\textrm{sinc}^2(at)\Longleftrightarrow\frac{1}{|a|}\textrm{tri}\left(\frac{f}{a}\right)\tag{3}$$

Equation $(3)$ is easily adapted to the equivalent non-unitary form of the Fourier transform with angular frequency $\omega=2\pi f$:

$$\textrm{sinc}^2(at)\Longleftrightarrow\frac{1}{|a|}\textrm{tri}\left(\frac{\omega}{2\pi a}\right)\tag{4}$$

It would be good if you made sure you're clear about all definitions mentioned above, and it would be even better if you sat down and calculated such a basic transform yourself in order to better understand what's going on instead of wasting time trying to make sense of seemingly contradictory information.

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