How does the Fourier transform work and why is it important?

[skaboy607]

Can anyone explain the above-i've read about in books, internet sites and still do not understand what its doing or the maths.

Thanks

 

[mathman]

Try wikipedia. It can't be explained in a few sentences.

 

[skaboy607]

hmm, I've read that and although i understand the basic, i.e complicated function and representing with smaller functions with sin and cosine waves, the rest doesn't make sense no matter how many times I read it.

 

[ice109]

how old are you? what level mathematical maturity do you have?

 

[skaboy607]

Not really sure why it matters but I am 21. hmmm not that much i guess?

 

[ice109]

you know anything about vectors? how you can express any vector as a sum of basis vectors?

 

[skaboy607]

hmmm can't say that I do, is that where I should start then?

 

[ice109]

what do you know then? why do you want to know about Fourier transforms?

 

[yyat]

An intuitive explanation:
I am sure you have seen one of these http://en.wikipedia.org/wiki/Spectral_analyzer" found on Hi-Fi's or digital audio players, that plot frequency vs. amplitude. They take the audio signal (amplitude/time), apply the (discrete) Fourier transform, and display the resulting function (amplitude/frequency).

To illustrate, take the function $$\cos(2\pi at)$$, which is a wave with frequency $$a$$. Its Fourier transform is zero except for two "spikes" at $$-a$$ and $$a$$.


A more mathematical reason why the Fourier transform is important, is that it turns differentiation into multiplication, see http://en.wikipedia.org/wiki/Fourier_Transform#Analysis_of_differential_equations", which is quite useful for solving some differential equations.