The Fourier transform and the spectrum

From time to frequencies

The Fourier transform generalizes the previous idea to all signals, periodic or not. It takes a signal described in time and returns its spectrum: a function that says, for each frequency, what amplitude (and what phase) it contributes to the signal. We move from the time domain to the frequency domain.

Reading a spectrum

A spectrum reads like the frequency ID card of the signal. A pure sinusoid gives a single line; a sum of sinusoids gives as many lines; noise spreads over all frequencies.

Time signal (left)                  Spectrum (right)

a pure sinusoid at 100 Hz           a single line at 100 Hz
  /\    /\    /\                        |
 /  \  /  \  /  \           <-->        |
     \/    \/                     ------+------------> frequency
                                      100 Hz

sum of 100 Hz and 300 Hz            two lines
  (more complex shape)      <-->      |     |
                                  ----+-----+--------> frequency
                                    100   300 Hz

Time and frequency: two views of the same object

The time signal and its spectrum contain exactly the same information, presented differently. The inverse transform lets you go back from the spectrum to the signal without losing anything. It is simply a change of viewpoint, like looking at an object from the front or from the side.

Applications everywhere

MP3 compression removes frequencies the ear perceives poorly, speech recognition analyzes the spectrum of the voice, medical imaging (MRI) reconstructs an image from frequency measurements. In all these cases, we work in the frequency domain because the problem becomes simpler there.

Classic pitfall

A thin, tall line in the spectrum does not mean that the frequency is "present for a long time": the amplitude measures the contribution of that frequency to the signal, not its duration. A brief, intense sound and a long, faint sound can occupy the same frequency with different amplitudes.