Understanding Discrete Fourier Transforms: A Guide for High School Students

[Ana09]

Fourier transforms. help!

hey! i am in high school, working on a project on discrete Fourier transforms..

I have the formula: X(k)= E^N-1_n=0x(n)(cos(wnk)-isin(wnk))
where w=2pi/N.

My question is: how do you express x(n)?does it have an equation, or what?
I really need help with this, so please if anyone knows.."
THANK YOU!

 

[FredGarvin]

What might be confusing you is the notation you have. x(n) implies x is a function of n. The actual notation is $$x_n$$ pronounced "x sub n." The $$x_n$$ are the individual numbers you are doing the DFT on.

http://en.wikipedia.org/wiki/Discrete_Fourier_transform

The sequence of N complex numbers x0, ..., xN−1 is transformed into the sequence of N complex numbers X0, ..., XN−1 by the DFT according to the formula

The actual DFT equation should be

$$X_k = \sum_{n=0}^{N-1} x_n e^{\frac{-2\pi i}{N}k n}$$

$$k=0,...N-1$$

 

[Ana09]

and what would those numbers be? the amplitude of each sample on the time domain?

anyway, thank you so much, it was really helpful =)

 

[Dale]

and what would those numbers be? the amplitude of each sample on the time domain?

Yes, exactly.

 

[FredGarvin]

If you tell us about your project it may be easier to help. It usually is just a long list of data points.

 

[Ana09]

Thanks!
Well, I kind of wanted to draw the spectra of different sounds "by hand". But I realized that with the DFT it would take far too long, so I was thinking on showing the beginning of the working, or an example of it, and then do the rest with the help of some software (which I am also trying to find online). Another option could be to draw the spectra with a really small number of samples, but I am not sure of whether the results would be realistic.

I have also read about FFT's, but I still don't really understand how they work. Would it be easier for me to use the FFT for this project?

 

[stabu]

My question is: how do you express x(n)?does it have an equation, or what?

The equation (a very standard one) gives X(n) in terms of x(n). Could it be that your teacher is simply looking for the inverse? i.e. x(n) in terms of X(n)? That's just the Inverse Fourier Transform then. Almost like changing places between X(n) and x(n) .. but not quite ... you'd better look it up to find it out exactly.

Usually the Discrete Fourier Transform is a numeric problem, it's very common to use computers ... there's are very many ways and packages ... it's a hugely useful operation, but not done with the head usually. The FFT is a very popular way of calculating it .. packages like Octave have it as standard. As you've given the equation .. it's in its very standard form, no specifics attached. It's all I can think of, that it must be the IDFT they want