Interpretation of the power spectrum after a discrete fourier transform?

Question

Suppose that you take a sequence of 100 numbers and transform them using the discrete fourier transform. You will get 51 complex values which can then be used to compute/plot the power spectrum. If a clear peak is visible in value (say) 20 (=0.2*N), does this mean that there is a periodic element repeated every 20 (in the original sequence) or a periodic element repeated ever 1/0.2=5 (again in the original sequence)??

I can see a visible peak at (say) 0.2*N in the power spectrum, but I am not sure what this tells me about the ORIGINAL sequence. Please explain...

Note: Please do not reply unless you really know this kind of stuff. I have already tried wikipedia and a few other sites and I need a clear and correct explanation.

Thank you!

Answer 1

A complex (but periodic) waveform is made up of a series (usually infinite in number) of α_n*e^(jwt) terms. Each of these terms (called 'Fourier components' of the original signal) contains some fraction of the original signals energy as given by the α_n (α sub n) complex coefficients. and the *total* power (summed over all of the Fourier component terms) is equal to the total power contained in the original signal.

The Fourier power spectrum simply tells you at which (complex) frequencies the most energy is found. Or, said in a slightly different way, how much energy is contained in each of the Fourier components.

Hope that helps.


Doug

Answer 2

I don't think so, because of the following reasoning.

Let's write P(x)= F(f(x)), where P is the power spectrum and F is the fourier transform. Irrespective of whether P is represented by discrete variable or continuous variable, the above relationship is still the same. When you talk about 20, or 0.2N etc. you are talking about the value of P at a single x value. All the peaks you talk about represent the same "x" value, only more refined at higher values of N. I mean as you add the value of N as it increases you make the value of P more accurate by adding higher dimentional corrections. Even though you may call them peaks, they are just refined corrections.

Answer 3

Yes I agree with your interpretation, there is an element repeating every 20. It may be buried in the noise, but it is there.