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I am confused in understanding use of fourier expansions of functions. This answer, for example says that we can write voice as a sum of sines and cosines of different frequencies and amplitudes, but how can we find those coefficients without knowing what exactly function is ? Also, when we record a voice, do we really know the function (like $f(x) = x^2-2x$) ? And if we know the function, then what is the need to find its series expansion.
In many books, exercises contain questions like to expand a given function to its fourier series. If we already know a function (and that too a simple function, like f(x) = x), then why would I want to expand it into a series ?

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Happy Mittal
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    Things like sound and image processing are done with sampled data, i.e. discrete samples of the function. You would therefore use a Discrete Fourier Transform (DFT), usually implement as the Fast Fourier Transform (FFT). – Andre Sep 22 '15 at 09:21
  • Ok, if I have discrete samples of a function, then why do I need to find it's DFT ? Also, in general, in real life, whenever we have to find fourier series of a function, then we have to have that function, and if we already have that function, then why do we need to find its fourier series ? – Happy Mittal Sep 22 '15 at 09:22
  • To do signal processing in frequency domain. Often to apply filters, like a Gaussian blur, for which you can make use of the convolution theorem. – Andre Sep 22 '15 at 09:24
  • You would usually be working with numerical data, not functions. Thus, finding the DFT would simply be a process of computing it. – AnonSubmitter85 Sep 27 '15 at 22:08

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