Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?
share|improve this question
    
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

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.