Fourier Transform

Written by Travis M. Moore
Last edited 26-Sep-2019

Time and Frequency Domains

The concept of time and frequency "domains" might sound intimidating, but it's really just describing two different ways to look at the same sound. In fact, we have already been viewing signals in both domains already. The time domain representation is simply a plot with amplitude on the y axis and time on the x axis. The frequency domain plots amplitude on the y axis and frequency on the x axis. Take a look at Figure 1 to refresh your memory.

Fast Fourier Transform
FIG. 1. © Travis M. Moore (2019)

The time domain view of a sine wave is the increases and decreases in amplitude over time: the typical shape of a "sine wave" we have come to know and love. What we don't know from the time domain plot, at least not from just a quick glance, is the frequency of the wave. (For the smarty-pants out there, yes, we could use the time axis to calculate the period of a cycle, then the frequency, but that's hardly information "at a glance.") The frequency view of a sine wave explicitly tells us what frequency (or frequencies) we're dealing with, but nothing about the duration of the sound.

So far we have been dealing with known frequencies, so we have just plotted in the time or frequency domain as necessary. However, what if we didn't know the frequency ahead of time? Or, given just the frequency, how would we be able to view the morphology of the wave over time? Well, a mathematician by the name of Joseph Fourier figured out how to do this, and he did so in the early 1800s, which means by hand. The concept is fairly simple, but extremely complex to carry out.

Moving from the time domain to the frequency domain is admittedly tricky. Luckily we do not need to concern ourselves with the math. All you need to know is that given any periodic (repeating) signal plotted as amplitude over time, the Fourier transform can figure out the sine waves (i.e., frequencies) needed to build (compose) that time waveform. This is called the Fourier Transform.

The reverse ("inverse") condition is moving from the frequency domain to the time domain. Appropriately, this is called the inverse Fourier transform. We have already seen sine waves combine to make a wave shaped like a square and a triangle. For example, dumping a bunch of odd harmonics (along with amplitudes!) into a Fourier transform is all that's needed to compose the square wave in time. You may recall learning about this process in the module on additive synthesis.

Test Your Understanding

The time domain is when you are viewing a signal's amplitude over time. Thus, amplitude is on the y axis and time is on the x axis. The signal will look like a sound wave.
The frequency domain is when you are viewing the individual sine wave components and amplitudes that make up a complex signal. Thus, amplitude is on the y axis and frequency is on the x axis. The signal will look like a set of lines, with position along the x axis indicating frequency and the height of the line indicating amplitude.
  • Fourier Transform: Time domain to frequency domain
  • Inverse Fourier Transform: Frequency domain to time domain


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