Fourier transform

• Mar 1st 2010, 06:05 AM
Niles_M
Fourier transform
Hi

I have a function in frequency-domain, which is almost periodic (it oscillates, but not in a very consistent way). My teacher told me that if I Fourier transform this function, then I will be able to extract the approximate period. Do you have any idea of what he is talking about?

Sincerely,
Niles.
• Mar 1st 2010, 01:17 PM
Laurent
Quote:

Originally Posted by Niles_M
I have a function in frequency-domain, which is almost periodic (it oscillates, but not in a very consistent way). My teacher told me that if I Fourier transform this function, then I will be able to extract the approximate period. Do you have any idea of what he is talking about?

Hi,
you can probably get the "approximate frequency" by looking at the frequency where the Fourier transform is maximum (in modulus).
Does it make sense?
• Mar 1st 2010, 10:04 PM
Niles_M
Lets take an example. If I Fourier Transform sin(x), then I get

$\displaystyle I \sqrt{\pi /2}\,\, \delta(-1 + \omega) - I \sqrt{\pi /2} \,\,\delta(1 + \omega)$

where delta is the Dirac delta. It is maximum in modulus for omega = -1 and omega=1. From this I should get 2Pi?

Thanks.
• Mar 2nd 2010, 01:53 AM
Laurent
Quote:

Originally Posted by Niles_M
Lets take an example. If I Fourier Transform sin(x), then I get

$\displaystyle I \sqrt{\pi /2}\,\, \delta(-1 + \omega) - I \sqrt{\pi /2} \,\,\delta(1 + \omega)$

where delta is the Dirac delta. It is maximum in modulus for omega = -1 and omega=1. From this I should get 2Pi?

Yes: the frequency with most contribution is indeed $\displaystyle f_0=1$, and the corresponding period is $\displaystyle \omega_0=\frac{2\pi}{f_0}$.

For a better example (one where there is not an exact periodicity), compute the Fourier transform of the function that equals $\displaystyle \sin x$ when $\displaystyle x\in[-a,a]$ and 0 elsewhere. This function is not perdiodic, hence you have no rigorous way to define a period. However, you'll see (computation is simple, if I remember well) that the Fourier transform is maximum at $\displaystyle 1$ (like above), as it "should be". I don't know if there are other simple examples, where computations can be done by hand, but you can try numerical experiments.
• Mar 2nd 2010, 01:57 AM
Niles_M
Thanky you very much. I really appreciate your help.

I have known about Fourier transformations for quite some time now, but it is the very first time that I read about these things. But it is interesting.

Again, thank you; it really means alot. For now I will work on some examples to get familiar with the concept. If there is any trouble, I will let you know.

Sincerely,
Niles.
• Mar 2nd 2010, 04:11 AM
Niles_M
Quote:

Originally Posted by Laurent
For a better example (one where there is not an exact periodicity), compute the Fourier transform of the function that equals $\displaystyle \sin x$ when $\displaystyle x\in[-a,a]$ and 0 elsewhere. This function is not perdiodic, hence you have no rigorous way to define a period. However, you'll see (computation is simple, if I remember well) that the Fourier transform is maximum at $\displaystyle 1$ (like above), as it "should be". I don't know if there are other simple examples, where computations can be done by hand, but you can try numerical experiments.

If e.g. I have a functions whose Fourier Transform has peaks at -4, -3, ..., 1, 2 etc., then is there a way to average over these values to find the most likely period of the function?
• Mar 2nd 2010, 09:27 AM
Laurent
Quote:

Originally Posted by Niles_M
If e.g. I have a functions whose Fourier Transform has peaks at -4, -3, ..., 1, 2 etc., then is there a way to average over these values to find the most likely period of the function?

Note that if $\displaystyle f$ is a frequency, then $\displaystyle 2f$ is also a frequency, etc, hence in your case the frequency is 1.
This case is no different from Fourier series: if a function is periodic with frequence $\displaystyle f_0$, its Fourier transform is a multiple of a sum of peaks at integer multiples of $\displaystyle f_0$.

In physics, the square modulus of the Fourier transform gives the "energy" of a given frequence. The fundamental frequency would be (loosely speaking) that with the most energy: either the highest peak (if sum of deltas) or the highest integral $\displaystyle \int_{f-\epsilon}^{f+\epsilon} |(\mathcal{F}\varphi)(\xi)|^2 d\xi$, which is to say, if $\displaystyle \mathcal{F}\varphi$ is continuous, the highest value of $\displaystyle |(\mathcal{F}\varphi)(\xi)|^2$.