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Fourier Transform equations

Ultimately, I want to understand spatial data operated on by Fourier Transforms, what am I loooking at in a transformed image. I did some introduction at uni and understand unit circles, Euler's Formula and linear combinations.

So far I am stuck trying to understand the equations in the image and want to be sure. Is the frequency a constant value in the integral of the time equation, and is time a constant for the integral of the frequency equation? If so, that would mean any integration is for a specific frequency or time.

So when a signal is broken into component periodic signals, how does it "know" the range of frequencies/time to transform? Or am I no where near understanding this?

THanks.

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If you know your integral transforms (cf. Laplace transform from any sophomore/junior level ODE class) then you know that the Fourier transform, as given by

$$ X(\Omega) = \int_\mathbb{R} x(t) e^{-j\Omega t} \ dt $$

returns a function of $\Omega$. That should be treated as "fixed" in the integrand. Similar reasoning holds for the inverse transform.

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  • $\begingroup$ So I was far off. Thanks. $\endgroup$ – user24007 May 26 '17 at 4:49

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