Joseph Fourier and the Fourier Transform
Joseph Fourier was a French scientist in the late 18th and early 19th centuries. He made important contributions to subjects ranging from algebra to thermodynamics, including early studies on the greenhouse effect on Earth’s climate, but today is best remembered for his discovery that many mathematical functions can be approximated more simply as a sum of basic trigonometric functions (sines and cosines).
This process is particularly useful to us because of the realization that you can analyze the structure of any waveform by breaking it down into a series of sine waves. By doing this, we can represent the wave as a list of simple sines and cosines, and their relative amplitudes and phases. We can build up a complex waveform by taking a single sine wave, then adding harmonics of it (sine waves whose frequency is an integral multiple of the fundamental sine wave) in different amplitudes and different phases.
We can then work with these sine and cosine waves mathematically in order to manipulate the original waveform. This is used in modern technology for many things, from audio equalizers on music players, to cleaning up errors in digital photographs, to analyzing the complex interference patterns from spectroscopy and crystallography used to identify substances in the laboratory.
This all sounds very complex; but the fundamentals of it are quite simple, and you can try it for yourself!
Our demonstration H4-04: Fourier Analysis can do this in the classroom, creating graphs like you see here. Each spike in the second graph represents one harmonic and its amplitude to make up the sawtooth wave in the first graph.
But you don't have to just take our word for it!