Fourier analysis

FourierFourier synthesisanalyse the output wave into its constituent harmonicscomplex sounds containing multiple tonesDiscriminant Fourier AnalysisFourier '''analysisFourier analysis and synthesisFourier analysis/techniqueFourier analyzerFourier component
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.wikipedia
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Fourier series

Fourier coefficientFourier expansionFourier coefficients
Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.
The process of deriving the weights that describe a given function is a form of Fourier analysis.

Trigonometric functions

cosinetrigonometric functiontangent
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.
They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.

Joseph Fourier

FourierJean Baptiste Joseph FourierJean-Baptiste Joseph Fourier
Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.
Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations.

Harmonic analysis

abstract harmonic analysisFourier theoryharmonic
Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis. The subsequent development of the field is known as harmonic analysis, and is also an early instance of representation theory.
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).

Discrete Fourier transform

DFTcircular convolution theoremFourier transform
That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.
The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications.

Partial differential equation

partial differential equationsPDEPDEs
Fourier analysis has many scientific applications – in physics, partial differential equations, 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.
An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.

Discrete-time Fourier transform

convolution theoremDFTDTFT § Properties
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.

Fast Fourier transform

FFTFast Fourier Transform (FFT)Fast Fourier Transforms
The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and important transformation on computers.
Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa.

Time–frequency analysis

time-frequency analysistime-frequency domainfrequency-time
As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these.
The practical motivation for time–frequency analysis is that classical Fourier analysis assumes that signals are infinite in time or periodic, while many signals in practice are of short duration, and change substantially over their duration.

Nuclear magnetic resonance

NMRmagnetic resonanceNuclear Magnetic Resonance (NMR)
CW spectroscopy is inefficient in comparison with Fourier analysis techniques (see below) since it probes the NMR response at individual frequencies or field strengths in succession.

Deferent and epicycle

epicyclesdeferentdeferents and epicycles
The classical Greek concepts of deferent and epicycle in the Ptolemaic system of astronomy were related to Fourier series (see Deferent and epicycle: Mathematical formalism).
Epicycles worked very well and were highly accurate, because, as Fourier analysis later showed, any smooth curve can be approximated to arbitrary accuracy with a sufficient number of epicycles.

Representation theory

linear representationrepresentationsrepresentation
The subsequent development of the field is known as harmonic analysis, and is also an early instance of representation theory.

List of Fourier-related transforms

Fourier-related transformFourier-related transforms
Each transform used for analysis (see list of Fourier-related transforms) has a corresponding inverse transform that can be used for synthesis.
This is a list of linear transformations of functions related to Fourier analysis.

Sine wave

sinusoidalsinusoidsine
That is, it takes a function from the time domain into the frequency domain; it is a decomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fourier transform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.
This property leads to its importance in Fourier analysis and makes it acoustically unique.

List of important publications in mathematics

Publications in topologyList of publications in mathematicsMémoire sur la propagation de la chaleur dans les corps solides
An early modern development toward Fourier analysis was the 1770 paper Réflexions sur la résolution algébrique des équations by Lagrange, which in the method of Lagrange resolvents used a complex Fourier decomposition to study the solution of a cubic:
Generally referred to simply as Tate's Thesis, Tate's Princeton Ph.D. thesis, under Emil Artin, is a reworking of Erich Hecke's theory of zeta- and L-functions in terms of Fourier analysis on the adeles.

Euler's formula

complex exponentialcomplex exponentialsEuler's exponential formula
can be represented as a recombination of complex exponentials of all possible frequencies:
In electrical engineering, signal processing, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see Fourier analysis), and these are more conveniently expressed as the sum of exponential functions with imaginary exponents, using Euler's formula.

Least-squares spectral analysis

Lomb periodogramLomb-ScargleLomb-Scargle periodogram
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis.

Spectral density

frequency spectrumpower spectrumspectrum
In signal processing, the Fourier transform often takes a time series or a function of continuous time, and maps it into a frequency spectrum.
According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range.

Babylonian mathematics

BabyloniansBabylonian mathematiciansBabylonian
A primitive form of harmonic series dates back to ancient Babylonian mathematics, where they were used to compute ephemerides (tables of astronomical positions).
They also used a form of Fourier analysis to compute ephemeris (tables of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.

Spectral density estimation

spectral estimationfrequency estimationspectral analysis
General mathematical techniques for analyzing non-periodic functions fall into the category of Fourier analysis.

Poisson summation formula

Poisson series
Parameter T corresponds to the sampling interval, and this Fourier series can now be recognized as a form of the Poisson summation formula.

Digital signal processing

DSPsignal processingdigital
. That is a cornerstone in the foundation of digital signal processing.

Basis (linear algebra)

basisbasis vectorbases
Hamel bases of spaces of this kind are typically not useful, whereas orthonormal bases of these spaces are essential in Fourier analysis.

Mathematics

mathematicalmathmathematician
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

Function (mathematics)

functionfunctionsmathematical function
In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.