Fourier transform

The Fourier transform, named after Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i.e. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes"). There are many closely-related variations of this transform, summarized below, depending upon the type of function being transformed. See also: List of Fourier-related transforms.

Applications

Fourier transforms have many scientific applications — in physics, number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, oceanography, optics, geometry, and other areas. (In signal processing and related fields, the Fourier transform is typically thought of as decomposing a signal into its component frequencies and their amplitudes.) This wide applicability stems from several useful properties of the transforms:

Related Topics:
Physics - Number theory - Combinatorics - Signal processing - Probability theory - Statistics - Cryptography - Acoustics - Oceanography - Optics - Geometry - Frequencies - Amplitude

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

• The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality).
• The transforms are invertible, and in fact the inverse transform has almost the same form as the forward transform.
• The sinusoidal basis functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. (For example, in a linear time-invariant physical system, frequency is a conserved quantity, so the behavior at each frequency can be solved independently.)
• By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.
• The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms.