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# What are the Fourier series and Fourier transforms?

The idea for the Fourier series and Fourier transforms was developed by French mathematician and physicist Baron Jean Baptiste Joseph Fourier (1768–1830). (For more about Fourier’s life and times, see above.)

A Fourier series was developed as an alternate method of expressing a function by the expansion of the function. It is actually a specific type of infinite mathematical series that involves trigonometric functions. More simply put, it is essentially an infinite sum of sine and cosine waves. The Fourier series is used in applied mathematics; the study of Fourier series is considered a branch of what is called Fourier analysis. In engineering and physics, it is used to split up periodic (or continuous) functions into a group of simpler terms; and in electronics, it is used to express the periodic functions seen in waveforms of communication signals. When Napoleon Bonaparte led his army into Egypt, French mathematician and physicist Baron Jean Baptiste Joseph Fourier joined him and was a scientific adviser. Fourier also worked for Napoleon when he was involved in draining the Bourgoin swamps and building a new highway.

The Fourier transform is a mathematical operation that essentially changes a signal into its constituent frequencies. For example, think of the Fourier transform as a “translator” that, as the name implies, “transforms” the signal. If one plots the electrical signal from a loudspeaker playing music, one sees a wavy (or sine) line, also referred to as the frequency. These waves can be big, small, close together, or farther apart. The collection of notes one hears from the speakers is the Fourier transform of the signal going into the loudspeaker. A type of Fourier transform is the fast Fourier transforms, or FFTs. They can be used for computer applications, with the speed and efficiency of the algorithm used to do the transform being “fast.”

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