From here: http://www.kvraudio.com/forum/viewtopic.php?p=989000&sid=6e7e5d21145a3efd03b8ebaa5a15f830#989000
The fourier transform converts a time representation ( samples ) into a frequency representation ( i.e. the frequency spectrum of these samples ) and vice versa.
Commonly it splits a number of audio samples, say 2048, into a corresponding number of sine waves of different frequencies, so that you know the volumes and the phase for each sine.
The principle is very simple:
If you multiply a set of samples with another set of reference samples and add each result, like this:
result = sample[ 0 ] * reference[ 0 ] + sample[ 1 ] * reference[ 1 ] + … + sample[ n ] * reference[ n ]
… the result is an indicator for the similarity of both sets of samples.
And now with sines
Now, if you want to know the volume and the phase of a sine at a certain frequency inside an audio sample, you have to calculate 2 of these similarity values:
FC = 2 * PI * frequency / samplerate; // “frequency constant”
result_i = sample[ 0 ] * sin( FC * 0.f ) + sample[ 1 ] * sin( FC * 1.f )+ … + sample[ n ] * sin( FC * n.f )
result_r = sample[ 0 ] * cos( FC * 0.f ) + sample[ 1 ] * cos( FC * 1.f )+ … + sample[ n ] * cos( FC * n.f )
To get the actual phase and magnitude, you use simple pythagoras to get length and trigonometry to get the angle:
magnitude = sqrt( result_i * result_i + result_r * result_r);
phase = atan2( result_i, result_r) // I think…
This way you can determine how much of a frequency is present in an audio file etc., and which phase it has relative to the beginning.
In Fourier Transform lingo, the sine part is oftenly called “imaginary” while the cosine part is called “rational”. They pretend to be complex numbers, but that’s more or less only a convention. The phase/magnitude representation IMHO feels more like complex numbers.
Getting the whole spectrum
Now, the idea behind fourier transforms is, if you start with a “fundamental” frequency of exactly 1 cycle over the whole length of the analysed set of samples (call it n samples) and do this with every multiple up to n/2 of the fundamental, you get a representation that thoroughly reflects the spectrum of the original sample. – If you transform it back (create a new bunch of n samples out of adding sines and cosines), it will perfectly fit the original samples!
Each “frequency slot” or multiple of the fundamental (1, 2, 3, …, n/2) is called a “bin”. (Note, there’s also a zero-bin, I’ll explain that later)
Fast Fourier Transform – FFT
This would be a very time consuming technique, if there wasn’t the Fast Fourier Transform, or FFT. The FFT is a simple, yet highly difficult to understand algorithm. It takes advantage of the fact, that if you draw all those sines in a graph, you have many lines that cross each other at certain points. And because they cross each other pretty often, you need only 1 multiply to gather information for many results needed. Hence, what FFT does is recursively reordering (“prepacking”, “butterfly”) the samples in a tricky way so that only a fraction of calculations are needed to create the spectrum for a given range of audio samples. The drawback is, FFT requires certain lengths (or “ns”) of samples, which are a power of 2: like 16, 32, 1024, 4096 samples etc.
DC offset and Nyquist in FFTs
Typically, a FFT outputs the spectrum in sines and cosines, but it also output 2 special values: A cosine representing the DC (offset of powers above zero and below zero), which is bin 0, or the fundamental frequency * 0 and a sine representing Nyquist (because at Nyquist, there can’t be a cosine!). Hence some algorithms require N/2+1 bins, others “pack” the Nyquist bin into the sine field of the DC bin.
Hope this helps, or maybe it’s a start…