Transforms are very useful in audio, but John Watkinson argues that they are just as useful for understanding audio as they are for processing it.
Let’s look first at the Fourier transform. The whole basis is that any periodic waveform can be broken down into a set of harmonically related pure tones, or sine waves. Please note the emphasis on periodic, because many types of sound are anything but periodic and we can come unglued if we don’t realise.
Essentially the Fourier Transform is a constructive use of aliasing. We generate a sine wave at a selected frequency, what we might call the probe signal, and multiply it by the waveform we want to analyse. Should that waveform contain a periodic component at the same frequency, then it will alias to zero Hz and the product waveform will have a finite average value which we call a coefficient. If the frequency is absent, the average will be zero.
Clearly if the probe signal is in quadrature with the signal we analyse, we won’t see the alias, and the obvious solution is to use two probe signals in quadrature: a sine and a cosine. Then we can see a periodic component of the signal having any phase, and the ratio of the coefficients preserves the phase.