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Mathematical transform that expresses a function of time as a function of frequency
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent
Fourier_transform
Discrete Fourier transform algorithm
Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT), or its inverse (IDFT), of a sequence. A Fourier transform
Fast_Fourier_transform
Function in discrete mathematics
In mathematics, the discrete Fourier transform (DFT) is a discrete version of the Fourier transform that converts a finite sequence of numbers into another
Discrete_Fourier_transform
Branch of mathematics
frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One can then re-synthesize the same sound
Fourier_analysis
Change of basis applied in quantum computing
the quantum Fourier transform (QFT) is a linear transformation on quantum bits, and is the quantum analogue of the discrete Fourier transform. The quantum
Quantum_Fourier_transform
Technique to analyze the infrared spectrum of matter
Fourier transform infrared spectroscopy (FTIR) is a technique used to obtain an infrared spectrum of absorption or emission of a solid, liquid, or gaseous
Fourier-transform infrared spectroscopy
Fourier-transform_infrared_spectroscopy
Fourier analysis technique applied to sequences
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values. The DTFT
Discrete-time Fourier transform
Discrete-time_Fourier_transform
Fourier-related transform for signals that change over time
The short-time Fourier transform (STFT) is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections
Short-time_Fourier_transform
Integral transform useful in probability theory, physics, and engineering
Laplace transform is related to many other transforms. It is essentially the same as the Mellin transform and is closely related to the Fourier transform. Unlike
Laplace_transform
Mathematical operation
fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to
Fractional_Fourier_transform
Variant Fourier transforms
In mathematics, the Fourier sine and cosine transforms are integral equations that decompose arbitrary functions into a sum of sine waves representing
Sine_and_cosine_transforms
Involutive change of basis in linear algebra
Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an
Hadamard_transform
Mathematical theorem about functions
mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively
Fourier_inversion_theorem
Spectroscopy based on time- or space-domain data
Fourier-transform spectroscopy (FTS) is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source
Fourier-transform spectroscopy
Fourier-transform_spectroscopy
Decomposition of periodic functions
Fourier transform Fast Fourier transform Fejér's theorem Fourier analysis Fourier inversion theorem Fourier sine and cosine series Fourier transform Gibbs
Fourier_series
Mathematical transform
classical Fourier transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. The Graph Fourier transform is
Graph_Fourier_transform
Discrete Fourier transform algorithm
The sparse Fourier transform (SFT) is a kind of discrete Fourier transform (DFT) for handling big data signals. Specifically, it is used in GPS synchronization
Sparse_Fourier_transform
Concept in applied mathematics
discrete Fourier transform (NUDFT or NDFT) of a signal is a type of Fourier transform, related to a discrete Fourier transform or discrete-time Fourier transform
Non-uniform discrete Fourier transform
Non-uniform_discrete_Fourier_transform
Generalization of the discrete Fourier transform
the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups. The Fourier transform
Fourier transform on finite groups
Fourier_transform_on_finite_groups
Generalisation of Fourier transform to any ring
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex
Discrete Fourier transform over a ring
Discrete_Fourier_transform_over_a_ring
arguments, Fourier-related transforms include: Two-sided Laplace transform Mellin transform, another closely related integral transform Laplace transform: the
List of Fourier-related transforms
List_of_Fourier-related_transforms
Mathematical operation
is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval
Hankel_transform
Signal representation
domains with a pair of mathematical operators called transforms. An example is the Fourier transform, which converts a time function into a complex valued
Frequency_domain
In mathematics the finite Fourier transform may refer to either another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g
Finite_Fourier_transform
Mathematical technique used in data compression and analysis
in time resolution at ascending frequencies for the Fourier transform and the wavelet transform is shown below. Note however, that the frequency resolution
Wavelet_transform
In algebraic geometry, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived
Fourier–Deligne_transform
In algebraic geometry, a Fourier–Mukai transform ΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which
Fourier–Mukai_transform
Short-time Fourier transform with variable resolution
constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier transform
Constant-Q_transform
Instrument in mass spectrometry
Fourier-transform ion cyclotron resonance mass spectrometry is a type of mass analyzer (or mass spectrometer) for determining the mass-to-charge ratio
Fourier-transform ion cyclotron resonance
Fourier-transform_ion_cyclotron_resonance
Linear transform from the time domain to the frequency domain
Laplace transform (the s-domain or s-plane). This similarity is explored in the theory of time-scale calculus. While the continuous-time Fourier transform is
Z-transform
Software library for computing discrete Fourier transforms
The Fastest Fourier Transform in the West (FFTW) is a software library for computing discrete Fourier transforms (DFTs) developed by Matteo Frigo and Steven
Fastest Fourier Transform in the West
Fastest_Fourier_Transform_in_the_West
French mathematician and physicist (1768–1830)
heat transfer and vibrations. The Fourier transform and Fourier's law of conduction are also named in his honour. Fourier is also generally credited with
Joseph_Fourier
fast Fourier transform is a type of fast Fourier transform algorithm over finite fields. This algorithm first decomposes a discrete Fourier transform into
Cyclotomic fast Fourier transform
Cyclotomic_fast_Fourier_transform
fast Fourier transform (HFFT) is a tool in image and signal processing which uses fast Fourier transform (FFT) routines to compute the discrete Fourier transform
Hexagonal fast Fourier transform
Hexagonal_fast_Fourier_transform
Integral transform closely related to the Fourier transform
mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions
Hartley_transform
Mathematical signal manipulation by computers
transform Fourier-related transforms Discrete Fourier transform Discrete-time Fourier transform Fast Fourier transform Short-time Fourier transform Filter
Digital_signal_processing
Multiresolution Fourier Transform is an integral fourier transform that represents a specific wavelet-like transform with a fully scalable modulated window
Multiresolution Fourier transform
Multiresolution_Fourier_transform
Integral transform in mathematics
The Radon transform is closely related to the Fourier transform. We define the univariate Fourier transform here as: f ^ ( ω ) = ∫ − ∞ ∞ f ( x ) e − 2 π
Radon_transform
Function for integral Fourier-like transform
The wavelets forming a continuous wavelet transform (CWT) are subject to the uncertainty principle of Fourier analysis respective sampling theory: given
Wavelet
Function acting on function spaces
integral operator (used to measure weighted shapes in the space). The Fourier transform is useful in applied mathematics, particularly physics and signal
Operator_(mathematics)
Mapping involving integration between function spaces
zeroes of the transform function. Note that there are alternative notations and conventions for the Fourier transform. Here integral transforms are defined
Integral_transform
Frequency of a chirp pulse
waveform, and the two versions are mathematically related by the Fourier transform. The spectrum is of particular interest when pulses are subject to
Chirp_spectrum
Mathematical function used in signal processing
⟷ Fourier transform sinc ( L f ) ≜ sin ( π L f ) π L f . {\displaystyle {\tfrac {1}{L}}\operatorname {rect} (x/L)\quad {\stackrel {\text{Fourier
Hann_function
Integral transform and linear operator
the sign of the frequency (see § Relationship with the Fourier transform). The Hilbert transform is important in signal processing, where it is a component
Hilbert_transform
Periodic distribution ("function") of "point-mass" Dirac delta sampling
framework of continuous Fourier analysis on tempered distributions, without any reference to Fourier series. The Fourier transform of a Dirac comb is another
Dirac_comb
Foundational principle in quantum physics
space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). A nonzero function and its Fourier transform cannot both
Uncertainty_principle
Conjecture about the behaviour of the Fourier transform on curved hypersurfaces
conjecture, also known as the Fourier restriction conjecture, is a conjecture about the behaviour of the Fourier transform on curved hypersurfaces. It was
Restriction_conjecture
quadratic Fourier transform is an integral transform that generalizes the fractional Fourier transform, which in turn generalizes the Fourier transform. Roughly
Quadratic_Fourier_transform
Duality for locally compact abelian groups
duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative
Pontryagin_duality
Area of mathematical analysis
arising from such decompositions. Basic examples include Fourier series and the Fourier transform, while modern harmonic analysis also studies maximal functions
Harmonic_analysis
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
rect ( x / a ) {\displaystyle \operatorname {rect} (x/a)} , its Fourier transform is ∫ − ∞ ∞ rect ( t a ) ⋅ e − i 2 π f t d t = a sin ( π a f )
Rectangular_function
Mathematical notation
(pronounced "v-hat"). This is especially common in physics context. The Fourier transform of a function f {\displaystyle f} is traditionally denoted by f ^
Hat_notation
Mathematical operation
to the Fourier transform) Widder, D. V. (1946), The Laplace Transform, Princeton University Press Elementary inversion of the Laplace transform. Bryan
Inverse_Laplace_transform
Error-correcting codes
R(x) = C(x) + E(x). Transform r(x) to R(x) using discrete Fourier transform. Since the calculation for a discrete Fourier transform is the same as the
Reed–Solomon_error_correction
Integral transform
space. The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name
Linear canonical transformation
Linear_canonical_transformation
Discrete fourier transform expressed as a matrix
mathematics, a DFT matrix is a square matrix as an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal
DFT_matrix
Indicator function of positive numbers
The Fourier transform of the Heaviside step function is a distribution. Using one choice of constants for the definition of the Fourier transform we have
Heaviside_step_function
Complex-valued function
transforms generalizing the Fourier transform, such that, for α = π / 2 {\displaystyle \alpha =\pi /2} , it reduces to the standard Fourier transform
Mehler_kernel
Algorithm to be run on quantum computers
quantum Fourier transform is the quantum analogue of the discrete Fourier transform, and is used in several quantum algorithms. The Hadamard transform is also
Quantum_algorithm
Algorithmic determination of wave cycle parts
his collaborators (see References). Here we consider 1-D discrete Fourier transform (DFT) phase retrieval problem. The DFT of a complex signal f [ n ]
Phase_retrieval
Theorem in mathematics
transform) it onto a (one-dimensional) line, and do a Fourier transform of that projection. Take that same function, but do a two-dimensional Fourier
Projection-slice_theorem
Indication of rate and sense of rotation
of negative frequency still applies. Fourier's original formulation (the sine transform and the cosine transform) requires an integral for the cosine
Negative_frequency
Signal processing conducted on analog signals
the Fourier transform integral is not used to determine the transform; instead, a table of transform pairs is used to find the Fourier transform of a
Analog_signal_processing
Number, approximately 3.14
to its own Fourier transform. Indeed, according to Howe (1980), the "whole business" of establishing the fundamental theorems of Fourier analysis reduces
Pi
Algorithm for phase retrieval
create computer-generated holograms. Let: FT – forward Fourier transform IFT – inverse Fourier transform i – the imaginary unit, √−1 (square root of −1) exp
Gerchberg–Saxton_algorithm
Method for solving certain nonlinear partial differential equations
In mathematics, the inverse scattering transform (or nonlinear Fourier transform) is a method that solves the initial value problem for a nonlinear partial
Inverse_scattering_transform
Theorem relating unitary operators to one-parameter Lie groups
Hamiltonian. Stone's Theorem can be recast using the language of the Fourier transform. The real line R {\displaystyle \mathbb {R} } is a locally compact
Stone's theorem on one-parameter unitary groups
Stone's_theorem_on_one-parameter_unitary_groups
Topics referred to by the same term
which Fourier inversion recovers a function from its Fourier transform Short-time Fourier transform or short-term Fourier transform (STFT), a Fourier transform
Fourier
Filter in electronics and signal processing
{a}{\pi }}}e^{-ax^{2}}} and the frequency response is given by the Fourier transform g ^ ( f ) = e − π 2 f 2 / a {\displaystyle {\hat {g}}(f)=e^{-\pi ^{2}f^{2}/a}}
Gaussian_filter
Common configuration for optical interferometry
operation of a Fourier transform spectrometer, which is essentially a Michelson interferometer with one movable mirror. (A practical Fourier transform spectrometer
Michelson_interferometer
Polynomial sequence
define powers, including fractional ones, of the Fourier transform, to wit a Fractional Fourier transform generalization, in effect a Mehler kernel. Folland
Hermite_polynomials
Mathematical operation
Laplace transform and closely related to the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of
Mellin_transform
Function used in signal processing
cosine transform. Two-dimensional windows are commonly used in image processing to reduce unwanted high-frequencies in the image Fourier transform. They
Window_function
Near-field diffraction
the others when the process leads to a known Fourier transform, and the connection with the Fourier transform is tightened in the linear canonical transformation
Fresnel_diffraction
Technique used in signal processing and data compression
a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are generally related to Fourier series
Discrete_cosine_transform
Mass spectrometry technique to induce fragmentation of selected ions in the gas phase
collision-induced dissociation (SORI-CID) is a CID technique used in Fourier transform ion cyclotron resonance mass spectrometry which involves accelerating
Collision-induced dissociation
Collision-induced_dissociation
Mathematical method in calculus
its Fourier transform decays at infinity at least as quickly as 1/|ξ|k. In particular, if k ≥ 2 {\displaystyle k\geq 2} then the Fourier transform is integrable
Integration_by_parts
Theorem in mathematics
mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a
Parseval's_theorem
Australian engineer
Systems 2000 CSIRO Medal for development and application of fast Fourier transform technology 1999–2001 Vice President Systems Engineering, Radiata Communications
John_O'Sullivan_(engineer)
Study of classical optics using Fourier transforms
Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination
Fourier_optics
Very general problem in computer science
representations of larger dimension for abelian groups. The quantum fourier transform can be defined in terms of Z N {\displaystyle \mathrm {Z} _{N}} ,
Hidden_subgroup_problem
Type of block code
Fourier transform can be described in a setting closer to the signal processing. Fourier transform over finite fields The discrete Fourier transform of
Cyclic_code
Mathematical algorithm
The chirp Z-transform (CZT) is a generalization of the discrete Fourier transform (DFT). While the DFT samples the Z plane at uniformly-spaced points
Chirp_Z-transform
Mathematical function common in physics
stretched exponential is also the characteristic function, basically the Fourier transform, of the Lévy symmetric alpha-stable distribution. In physics, the
Stretched exponential function
Stretched_exponential_function
Analytical technique based on determining mass to charge ratio of ions
spectrometry (e.g. product ion scans). Fourier-transform mass spectrometry (FTMS), or more precisely Fourier-transform ion cyclotron resonance mass spectrometry
Mass_spectrometry
Recursive algorithm in applied mathematics
In applied mathematics, the sliding discrete Fourier transform is a recursive algorithm to compute successive STFTs of input data frames that are a single
Sliding_DFT
Cryptographic hash function
provably secure hash functions. It is based on the concept of the fast Fourier transform (FFT). SWIFFT is not the first hash function based on the FFT, but
SWIFFT
Wave shaped like the sine function
Fourier series are frequently used in signal processing and the statistical analysis of time series. The Fourier transform (FT) then extended Fourier
Sine_wave
Relative importance of certain frequencies in a composite signal
from time series data such as these involves the Fourier transform, and generalizations based on Fourier analysis. In many cases the time domain is not
Spectral_density
Reconstruction of a filtered signal
{\displaystyle F=H/G\,} where F is the estimated Fourier transform of f. Finally, the inverse Fourier transform of the function F is taken to find the estimated
Deconvolution
Mathematical analysis of frequency content of signals
more dimensions. One of the more popular multidimensional transforms is the Fourier transform, which converts a signal from a time/space domain representation
Multidimensional_transform
Mathematical theorem
of a function or distribution at infinity with analyticity of its Fourier transform. It is named after Raymond Paley (1907–1933) and Norbert Wiener (1894–1964)
Paley–Wiener_theorem
Integral transform
|}}\,\mathrm {d} \omega } is admissible constant, where hat means Fourier transform operator. Sometimes, ψ ~ ( t ) = ψ ( t ) {\displaystyle {\tilde {\psi
Continuous_wavelet_transform
Integral expressing the amount of overlap of one function as it is shifted over another
needed] For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution
Convolution
of Fourier analysis topics and list of Fourier-related transforms, which are more directed towards the classical Fourier series and Fourier transform of
List of harmonic analysis topics
List_of_harmonic_analysis_topics
representation with perfect time resolution. In contrast, the magnitude of the Fourier transform (FT) of the signal may be considered as a representation with perfect
Time–frequency_representation
Systematic collection of geophysical data for spatial studies
wave. The motivation for development of the Wavelet transform was the Short-time Fourier transform. The signal to be analysed, say f(t) is multiplied with
Geophysical_survey
Tent function, often used in signal processing
\end{cases}}\end{aligned}}} The transform is easily determined using the convolution property of Fourier transforms and the Fourier transform of the rectangular function:
Triangular_function
Mathematical transform
In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical
Fourier–Bros–Iagolnitzer transform
Fourier–Bros–Iagolnitzer_transform
Special case of the short-time Fourier transform
The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform. It is used to determine the sinusoidal frequency
Gabor_transform
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