fft - What does the exponential term in the Fourier transform mean?

We know that Fourier transform $F(\omega)$ of function $f(t)$ is summation from $-\infty$ to $+\infty$ product of $f(t)$ and $e^{-j \omega t}$:

$$F(\omega) = \int\limits_{-\infty}^{+\infty} f(t) \ e^{-j \omega t} \ dt$$

Here, what does the exponential term mean?

Answer

It's a complex exponential that rotates forever on the complex plane unit circle:

$$e^{-j\omega t} = \cos(\omega t) + j \sin(\omega t).$$

You can think of Fourier transform as calculating correlation between $f(t)$ and a complex exponential of each frequency, comparing how similar they are. Complex exponentials like that have the nice quality that they can be time-shifted by multiplying them with a complex number of unit magnitude (a constant complex exponential). If the Fourier transform result at a particular frequency is a non-real complex number, then the complex exponential of that frequency can be multiplied by that complex number to get it shifted in time so that the correlation to $f(t)$ is maximized.