A mathematical technique that provides a way to analyze the frequency content of a waveform. More generally it allows any math function to be transferred from one domain to another. To better understand this it's important to note that all audio waveforms can be represented by mathematical equations or functions. Fourier analysis, which was developed by Baron Fourier in the 19th century (while working on problems of heat transfer in artillery for Napoleon), is the math required to go back and forth between domains such as time and frequency. For example, an audio waveform can be transformed from its normal domain of time (meaning we normally look at audio waveforms as a change of energy over time - this is where we see the "shape" of a waveform) to the domain of frequency, where we can look at the harmonic content of a waveform broken down into frequencies (but without any measure of time).
Fourier determined that the all waveforms are composed of multiple sine waves summed together. A Square wave, for example, is composed of a series of sine waves at systematic frequencies. Any waveform can be divided into its foundational components - a series of sine waves at differing frequencies, amplitudes and phase relationships. The Fourier Transform is a way of taking a waveform and determining the sine waves that create it.
A Fourier transform is both the name of the plot that tells us the frequency content of a waveform and is also the name of the mathematical formula that determines it from a given waveform.
In audio, a Fourier transform assigns the various frequencies or periodicities present in a complex waveform to specific frequencies in the spectrum. For example, a sine wave at 500 Hz repeats itself 500 times per second when looked at in the time domain. When looked at in the frequency domain we can simply see that there is one frequency component: 500 Hz. A 500 Hz square wave, on the other hand, looks different in the time domain - a square wave as opposed to the rolling curves of a sine wave - and when looked at in the frequency domain we can see that a square wave of a given frequency is produced by having all the odd harmonics present and in phase, but at increasingly lower levels as frequency rises. In our 500 Hz example there would be a harmonic at 1500 Hz, but lower in amplitude than the fundamental of 500 Hz. To be clear, this is a sine wave at this frequency. Then there is another sine wave at 2500 Hz that's even lower in level, and so on. This combination of sine waves is what makes up a square wave. Applying a Fourier Transform to a square wave would give us a frequency plot that might look something like this:
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Fundamental 1st odd harmonic 2nd odd harmonic etc.
This particular graph only shows frequencies and their relative amplitudes. It is a graph of the spectra or the spectral (harmonic) content of a waveform. Other information such as phase is also separated when Fourier Transforms are done, but these aren't as widely used and are beyond the scope of this writing.
The reason why Fourier Transforms are important in audio is they allow us to clearly see the frequency content present in a particular audio waveform. They also happen to mimic the way our ears hear sound. We tend to analyze sounds in terms of how they relate to pitch. Looking at a complex waveform on an oscilloscope generally tells us very little about what it might sound like, while looking at it in terms of its frequency components can tell us a lot, especially as we correlate these frequencies to musical pitches.
Fourier analysis has been and continues to be widely used in the audio industry and has far-reaching consequences, especially with digital audio, which is largely based on being able to perform mathematical operations with numbers that represent sound. FFT (short for Fast Fourier Analysis) - a term that is thrown around in the audio industry quite a bit - is a process derived from the above. We'll cover this in more detail later.
