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Does anyone have a mechanism to understand intuitively ( and automatically ) why the fourier transform of certain functions have certain shapes ( at least for some functions, not necessarily for all ) ? I know what kind of operator the fourier transform is and what it does to a function but somehow i can't see intuitively and automatically why why the fourier transform of certain functions have certain shapes. For example , is there a intuitive reason the fourier transform of a pulse ( box function ) is in the sync shape ?
Thanks in advance

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I am not sure about intuition in general, but regarding the step-function FT being a sync function:

Note that the shape will remain the same, but the frequencies over which the FT of a particular step-function resides is a function of the pulse-width of the original signal. Namely, expanding a function in the time-domain actually shrinks the corresponding frequency-domain function (think slowing down voice recordings, the sound gets very low i.e. lower frequency).

That being said, as you decrease the pulse-width of a particular step-function the frequency components of that signal increase because now there is more change happening (to use a loose descriptor) in a shorter amount of time.

In contrast, if we expand the step-function in the time-domain to have a longer pulse-width then there is less change and the corresponding frequency components must be much lower.

In general, I look at a function and try and get a feel for how quickly it might be changing to get a rough idea. But as I said, I don't know of any general rule of thumb here.

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