What’s a Fourier transform?

A Fourier transform is a mathematical operation used in various fields of science and engineering to analyze and manipulate signals, functions, or data in the frequency domain. It is named after the French mathematician and physicist Jean-Baptiste Joseph Fourier, who made significant contributions to the study of heat transfer and the mathematics of periodic functions.

The Fourier transform takes a time-domain signal or function and decomposes it into its constituent frequencies. In other words, it converts a signal from the time or spatial domain into the frequency domain. This transformation allows us to analyze the signal's frequency components, making it useful in fields such as signal processing, image processing, audio analysis, and many other areas.

Mathematically, the continuous Fourier transform of a function f(t) is given by:

F(ω) = ∫[from -∞ to +∞] f(t) * e^(-jωt) dt

Here:

F(ω) is the complex function in the frequency domain.
f(t) is the function in the time domain.
ω represents the angular frequency (2π times the frequency of the sinusoid).
j is the imaginary unit (√(-1)).