Editing Fourier analysis (section)

==Applications==
Fourier analysis has many scientific applications – in [[physics]], [[partial differential equation]]s, [[number theory]], [[combinatorics]], [[signal processing]], [[digital image processing]], [[probability theory]], [[statistics]], [[forensics]], [[option pricing]], [[cryptography]], [[numerical analysis]], [[acoustics]], [[oceanography]], [[sonar]], [[optics]], [[diffraction]], [[geometry]], [[protein]] structure analysis, and other areas.

This wide applicability stems from many useful properties of the transforms''':'''
* The transforms are [[linear operator]]s and, with proper normalization, are [[unitary operator|unitary]] as well (a property known as [[Parseval's theorem]] or, more generally, as the [[Plancherel theorem]], and most generally via [[Pontryagin duality]]).<ref name=Rudin/>
* The transforms are usually invertible.
* The [[exponential function]]s are [[eigenfunction]]s of [[derivative|differentiation]], which means that this representation transforms linear [[differential equation]]s with [[constant coefficients]] into ordinary algebraic ones.<ref name=Evans/> Therefore, the behavior of a [[LTI system|linear time-invariant system]] can be analyzed at each frequency 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 signal filtering, [[polynomial]] multiplication, and [[Multiplication algorithm#Fourier transform methods|multiplying large numbers]].<ref name=Knuth/>
* The [[Discrete Fourier transform|discrete]] version of the Fourier transform (see below) can be evaluated quickly on computers using [[fast Fourier transform]] (FFT) algorithms.<ref name=Conte/>

In forensics, laboratory infrared spectrophotometers use Fourier transform analysis for measuring the wavelengths of light at which a material will absorb in the infrared spectrum. The FT method is used to decode the measured signals and record the wavelength data. And by using a computer, these Fourier calculations are rapidly carried out, so that in a matter of seconds, a computer-operated FT-IR instrument can produce an infrared absorption pattern comparable to that of a prism instrument.<ref name=Saferstein/>

Fourier transformation is also useful as a compact representation of a signal. For example, [[JPEG]] compression uses a variant of the Fourier transformation ([[discrete cosine transform]]) of small square pieces of a digital image. The Fourier components of each square are rounded to lower [[precision (arithmetic)|arithmetic precision]], and weak components are eliminated, so that the remaining components can be stored very compactly. In image reconstruction, each image square is reassembled from the preserved approximate Fourier-transformed components, which are then inverse-transformed to produce an approximation of the original image.

In [[signal processing]], the Fourier transform often takes a [[time series]] or a function of [[continuous time]], and maps it into a [[frequency spectrum]]. That is, it takes a function from the time domain into the [[frequency]] domain; it is a [[orthogonal system|decomposition]] of a function into [[Sine wave|sinusoids]] of different frequencies; in the case of a [[Fourier series]] or [[discrete Fourier transform]], the sinusoids are [[harmonic]]s of the fundamental frequency of the function being analyzed.

When a function <math>s(t)</math> is a function of time and represents a physical [[Signal (information theory)|signal]], the transform has a standard interpretation as the frequency spectrum of the signal. The [[magnitude (mathematics)|magnitude]] of the resulting complex-valued function <math>S(f)</math> at frequency <math>f</math> represents the [[amplitude]] of a frequency component whose [[phase (waves)|initial phase]] is given by the angle of <math>S(f)</math> (polar coordinates).

Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied to analyze ''spatial'' frequencies, and indeed for nearly any function domain. This justifies their use in such diverse branches as [[image processing]], [[heat conduction]], and [[automatic control]].

When processing signals, such as [[Sound|audio]], [[radio wave]]s, light waves, [[seismic wave]]s, and even images, Fourier analysis can isolate narrowband components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.<ref name=Rabiner/>

Some examples include''':'''
* [[Equalization (audio)|Equalization]] of audio recordings with a series of [[bandpass filter]]s;
* Digital radio reception without a [[superheterodyne]] circuit, as in a modern cell phone or [[radio scanner]];
* [[Image processing]] to remove periodic or [[anisotropic]] artifacts such as [[jaggies]] from [[interlaced video]], strip artifacts from [[strip aerial photography]], or wave patterns from [[radio frequency interference]] in a digital camera;
* [[Cross correlation]] of similar images for co-alignment;
* [[X-ray crystallography]] to reconstruct a crystal structure from its diffraction pattern;
* [[Fourier-transform ion cyclotron resonance]] mass spectrometry to determine the mass of ions from the frequency of cyclotron motion in a magnetic field;
* Many other forms of spectroscopy, including [[infrared spectroscopy|infrared]] and [[nuclear magnetic resonance]] spectroscopies;
* Generation of sound [[spectrogram]]s used to analyze sounds;
* Passive [[sonar]] used to classify targets based on machinery noise.