Interpretation of the Fourier coefficients?

Suppose I have a discrete function $f( x_i ) = y_i$.

I can use these pairs $(x_i, y_i)$ as complex number $z_i = x_i + j \, y_i$.

Now, having this set $z_i$, I can apply discrete Fourier transform, as show in Wikipedia.

Now, suppose the calculated Fourier coefficients are ${X_i}$, where each ${X_i}$ is, of course, a complex number.

So, what is the interpretation of these numbers? For example:

• what does the real part of these number means (if anything at all)?
• what does the imaginary part of these number means (if anything at all)?
• what does the module $|{X_i}|$ means? Does these values give the spectrum of the function? If so, what it's used for?
-
This makes little sense to me. Some motivation? – leonbloy Jul 5 '12 at 20:25
@leonbloy - Just out of curiosity. – Kiril Kirov Jul 5 '12 at 20:27

This makes little sense to me in the original setting, i.e., if we regard the points as $(x,y)$ values of a function. For one thing, a mere permutation of the points (which is not significant) gives different Fourier transforms.
But if you regard your input $\{ (x_1,y_1), (x_2,y_2) \cdots\}$, not only as a set of points belonging to the graph of the function $f(x)$, but rather as true sequence (a list, where order is significant) which travels along that graph (like succesive points along a parametric curve), then it makes sense and can be quite useful: see for example Fourier descriptors. Of course, in this case we don't need to restrict to true functions, we just deal with general parametric curves in the $(x,y)$ plane.
Ha, this is interesting, I didn't know, that $\{ (x_1,y_1), (x_2,y_2) \cdots\}$ could be "random" points in the (x, y) plane. Thanks for the example with Fourier descriptors, this is really interesting :) +1 and accepted. By the way, I'd be happy to see some more examples. Thanks! – Kiril Kirov Jul 6 '12 at 9:16