How does the Fourier transform get you the frequency amplitude I understand that the Fourier transforn gets you the function which gives the amplitude of each frequency. But I don't understand how that is possible by multiplying it by an exponential. How is that possible?
EDIT: Since my question seems to be unclear: how do you get the amplitude of a frequency f, by integrating it and multiplying it with exp(-i*2*Pi*f) ?
 A: Let's look at the equation of the Inverse Fourier Transform.
$$f(x)=\int_{-\infty}^{\infty}F(s)\text{e}^{i2\pi sx}ds$$
The integral is "like" a summation, so let's replace it with a summation to make it easier.
$$f(x)=\sum_{s=-\infty}^{\infty}F(s)\text{e}^{i2\pi sx}\Delta s$$
We've approximated the integration with a Riemann sum.  This is not necessary, but maybe it makes things a bit easier to see.
We can rewrite this using Euler's identity.
$$f(x)=\sum_{s=-\infty}^{\infty}F(s)\left(\cos(2\pi sx)+i\sin(2\pi sx)\right)\Delta s$$
So we see that our function $f$ is really just a sum of a bunch of sinusoids.  What are the frequencies of these sinusoids?  And what are their amplitudes?  Recall that the function $g(x)=A\cos(\omega x)$ has frequency $\omega$ with amplitude $A$
So we see that the frequencies of the sinusoids are $2\pi s$ and the amplitudes are $F(s)$.
$F$ is often called the spectrum of the function $f$.
A: So what is the definition of the Fourier Transform?  The definition is as follows:
$$F(s)=\int_{-\infty}^{\infty}f(x)\text{e}^{-i2\pi sx}dx$$
What is the definition of the inverse Fourier Transform?  It's very similar, and is as follows:
$$f(x)=\int_{-\infty}^{\infty}F(s)\text{e}^{i2\pi sx}ds$$
Note that there are other equivalent definitions, but we will work with these.
So where does frequency come into the picture?  People often call $s$ the frequency.  Integration is "like" a summation.  Euler's identify is also relevant: $\text{e}^{ia}=\cos(a) + i\sin(a)$.  So $\text{e}^{i2\pi sx}=\cos(2\pi sx)+i\sin(2\pi sx)$.  So $2\pi s$ is the frequency of the co/sinusoid.  And what is the amplitude of that sinusoid? It's $F(s)$.
The function $F$ is often called the "spectrum" of $f$.
Does this help?
