A fourier transform decomposes an arbitrary signal in time space to frequency space. Ok ok i know this is the vague textbook definition, but bear with me. imagine any wave you see in your life - for example on a music player visualizer. (Those little waves that jump according to how the music changes.) Fourier said that this complex looking wave can be decomposed into many individual sine waves with different frequencies. In other words, that complicated wave is just many simple looking sine waves added together. A fourier transform takes the crazy wave and spits out a list of the different frequency sine waves that make it up.
Remember:
A * sin(B *t)
A = magnitude
B = frequency
t indicates time "space"
ok i am in wayyyyyy over my head but maybe my input can be of use
before the popularity of the PC, personal computer..not political correct crap we now have
i wss product manager of a metrology product line ,,surface texture measurement, used in industry to qualify a machined surface
back then all we had was a gage head with diamond stylus..the stylus moved up and down relative to the machined surface that it traversed and generated a signal that was output to an analog strip chart recorder..all we had wa a squiggly line on the paper
so how in the world do you describe a squiggly line?
well, Fourier Analysis wa a mathematical represntation of fucntions as linear combinations of sine ,cosine or complex exponential harmonic components..it defines frequency of the wave
this was not enuff so we looked at DFT or Discrete Fourier Transform - the mathematical calculation of the RMS order of magnitudeof the wave..ie. gives amplitude height
still not enuf to diferentiate one squiggly line from another so we used FFT or Fast Fourier Transformation which gave us a Harmonic Analysis of the line..we could then compare the lines without all the subjective second guessing ...
ther are a whole lot of people on this forum who know way more about this
my point is to show practical use of FFT etc..
Numerically (think CFD), we like Fourier transforms because we know how to calculate derivatives of sines. Due to discretization, we cannot calculate derivatives of functions that we dont' have enough points per wavelength across (Nyquist limit). This lets us perform error analysis on derivatives and get an idea how much grid we need for a specific problem, what waves we're calculation correctly, and lots of other goodies.
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