In Section 2.2.3, we showed how sound can be represented graphically in two ways.  In the waveform view, time is on the horizontal axis and amplitude of the sound wave is on the vertical axis.  In the frequency analysis view, frequency is on the horizontal axis and the magnitude of the frequency component is on the vertical axis.  The waveform view represents sound in the time domain. The frequency analysis view represents sound in the frequency domain.  (See Figure 2.18 and Figure 2.19.)  Whether sound is represented in the time or the frequency domain, it's just a list of numbers.  The information is essentially the same – it's just that the way we look at it is different.

The one-dimensional Fourier transform is function that maps real to the complex numbers, given by the equation below.  It can be used to transform audio data from the time to the frequency domain.

F\left ( n \right )=\int_{-\infty }^{\infty }f\left ( k \right )e^{-i2\pi nk}dk\: where\: i=\sqrt{-1}

Equation 2.12 Fourier transform (continuous)

Sometimes it's more convenient to represent sound data one way as opposed to another because it's easier to manipulate it in a certain domain.  For example, in the time domain we can easily change the amplitude of the sound by multiplying each amplitude by a number.  On the other hand, it may be easier to eliminate certain frequencies or change the relative magnitudes of frequencies if we have the data represented in the frequency domain.