# Category: Fourier analysis

Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perc
Metaplectic group
In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an a
Window function
In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, norm
Reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the
Discrete Fourier transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fouri
Fourier–Bros–Iagolnitzer transform
In mathematics, the FBI transform or Fourier–Bros–Iagolnitzer transform is a generalization of the Fourier transform developed by the French mathematical physicists Jacques Bros and Daniel Iagolnitzer
Indirect Fourier transform
In a Fourier transform (FT), the Fourier transformed function is obtained from by: where is defined as . can be obtained from by inverse FT: and are inverse variables, e.g. frequency and time. Obtaini
Hexagonal Efficient Coordinate System
The Hexagonal Efficient Coordinate System (HECS), formerly known as Array Set Addressing (ASA), is a coordinate system for hexagonal grids that allows hexagonally sampled images to be efficiently stor
Spectral concentration problem
The spectral concentration problem in Fourier analysis refers to finding a time sequence of a given length whose discrete Fourier transform is maximally localized on a given frequency interval, as mea
Frequency selective surface
A frequency-selective surface (FSS) is any thin, repetitive surface (such as the screen on a microwave oven) designed to reflect, transmit or absorb electromagnetic fields based on the frequency of th
Von Neumann stability analysis
In numerical analysis, von Neumann stability analysis (also known as Fourier stability analysis) is a procedure used to check the stability of finite difference schemes as applied to linear partial di
Periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of radians, are periodic functions. Periodic functio
Sine and cosine transforms
In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fo
Spectral component
In telecommunications, spectral component is any of the waves that range outside the interval of assigned to a signal. Any waveform can be disassembled into its spectral components by Fourier analysis
Spherical harmonics
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific
Set of uniqueness
In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis.
Bispectrum
In mathematics, in the area of statistical analysis, the bispectrum is a statistic used to search for nonlinear interactions.
Chirplet transform
In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets. Similar to the wavelet transform, chirplets are usually gener
Interpolation space
In the field of mathematical analysis, an interpolation space is a space which lies "in between" two other Banach spaces. The main applications are in Sobolev spaces, where spaces of functions that ha
Microlocal analysis
In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear parti
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for whose vectors are orthonormal, that is, they are all unit vectors an
In signal processing, the overlap–add method is an efficient way to evaluate the discrete convolution of a very long signal with a finite impulse response (FIR) filter : where h[m] = 0 for m outside t
Superoscillation
Superoscillation is a phenomenon in which a signal which is globally band-limited can contain local segments that oscillate faster than its fastest Fourier components. The idea is originally attribute
Autocovariance
In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is close
Spectrum continuation analysis
Spectrum continuation analysis (SCA) is a generalization of the concept of Fourier series to non-periodic functions of which only a fragment has been sampled in the time domain. Recall that a Fourier
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , (wh
Beevers–Lipson strip
Beevers–Lipson strips were a computational aid for early crystallographers in calculating Fourier transforms to determine the structure of crystals from crystallographic data, enabling the creation of
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on
Single-shot multi-contrast X-ray imaging
Single-shot multi-contrast x-ray imaging is an efficient and a robust x-ray imaging technique which is used to obtain three different and complementary types of information, i.e. absorption, scatterin
Almost periodic function
In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-pe
In financial mathematics, the Carr–Madan formula of Peter Carr and Dilip B. Madan shows that the analytical solution of the European option price can be obtained once the explicit form of the characte
DFT matrix
In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication.
Periodogram
In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated meth
Fraunhofer diffraction equation
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is view
Tube domain
In mathematics, a tube domain is a generalization of the notion of a vertical strip (or half-plane) in the complex plane to several complex variables. A strip can be thought of as the collection of co
Wirtinger's inequality for functions
In mathematics, historically Wirtinger's inequality for real functions was an inequality used in Fourier analysis. It was named after Wilhelm Wirtinger. It was used in 1904 to prove the isoperimetric
Motions in the time-frequency distribution
Several techniques can be used to move signals in the time-frequency distribution. Similar to computer graphic techniques, signals can be subjected to horizontal shifting, vertical shifting, dilation
Homogeneous distribution
In mathematics, a homogeneous distribution is a distribution S on Euclidean space Rn or Rn \ {0} that is homogeneous in the sense that, roughly speaking, for all t > 0. More precisely, let be the scal
Photon Doppler velocimetry
Photon Doppler velocimetry (PDV) is a one-dimensional Fourier transform analysis of a heterodyne laser interferometry, used in the shock physics community to measure velocities in dynamic experiments
Modified discrete cosine transform
The modified discrete cosine transform (MDCT) is a transform based on the type-IV discrete cosine transform (DCT-IV), with the additional property of being lapped: it is designed to be performed on co
SigSpec
SigSpec (acronym of SIGnificance SPECtrum) is a statistical technique to provide the reliability of periodicities in a measured (noisy and not necessarily equidistant) time series. It relies on the am
Hartley transform
In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed
Fractional Fourier transform
In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier
Rigorous coupled-wave analysis
Rigorous coupled-wave analysis (RCWA) is a semi-analytical method in computational electromagnetics that is most typically applied to solve scattering from periodic dielectric structures. It is a Four
Laplace's equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as or where is
Ewald summation
Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g. electrostatic interactions) in periodic systems. It was first developed as the method for calcula
Parametrix
In mathematics, and specifically the field of partial differential equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a
Stationary-wave integrated Fourier-transform spectrometry
Stationary-wave integrated Fourier-transform spectrometry (SWIFTS), or standing-wave integrated Fourier-transform spectrometry, is an analytical technique used for measuring the distribution of light
Non-uniform discrete Fourier transform
In applied mathematics, the nonuniform 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
Conjugate Fourier series
In the mathematical field of Fourier analysis, the conjugate Fourier series arises by realizing the Fourier series formally as the boundary values of the real part of a holomorphic function on the uni
Discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function.
Generalized Fourier series
In mathematical analysis, many generalizations of Fourier series have proved to be useful. They are all special cases of decompositions over an orthonormal basis of an inner product space. Here we con
Gabor atom
In applied mathematics, Gabor atoms, or Gabor functions, are functions used in the analysis proposed by Dennis Gabor in 1946 in which a family of functions is built from translations and modulations o
Schwartz space
In mathematics, Schwartz space is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on t
Oscillatory integral operator
In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form where the function S(x,y) is called the phase of the operator and the function a
Fourier-transform spectroscopy
Fourier-transform spectroscopy is a measurement technique whereby spectra are collected based on measurements of the coherence of a radiative source, using time-domain or space-domain measurements of
Littlewood–Paley theory
In harmonic analysis, a field within mathematics, Littlewood–Paley theory is a theoretical framework used to extend certain results about L2 functions to Lp functions for 1 < p < ∞. It is typically us
Spin-weighted spherical harmonics
In special functions, a topic in mathematics, spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphe
Spectral leakage
The Fourier transform of a function of time, s(t), is a complex-valued function of frequency, S(f), often referred to as a frequency spectrum. Any linear time-invariant operation on s(t) produces a ne
Whittaker–Shannon interpolation formula
The Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real numbers. The formula dates back to the works o
Modulus of continuity
In mathematical analysis, a modulus of continuity is a function ω : [0, ∞] → [0, ∞] used to measure quantitatively the uniform continuity of functions. So, a function f : I → R admits ω as a modulus o
Quantum Fourier transform
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 Fourier transform is a
Poisson kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit d
Pontryagin duality
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative
S transform
S transform as a time–frequency distribution was developed in 1994 for analyzing geophysics data. In this way, the S transform is a generalization of the short-time Fourier transform (STFT), extending
Hausdorff–Young inequality
The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by William Henry Young and extended by Hau
Convolution power
In mathematics, the convolution power is the n-fold iteration of the convolution with itself. Thus if is a function on Euclidean space Rd and is a natural number, then the convolution power is defined
Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential equation for an arbitrary complex
Unit impulse
No description available.
Regressive discrete Fourier series
In applied mathematics, the regressive discrete Fourier series (RDFS) is a generalization of the discrete Fourier transform where the Fourier series coefficients are computed in a least squares sense
Fourier analysis
In mathematics, Fourier analysis (/ˈfʊrieɪ, -iər/) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from t
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives
Optical correlator
An optical correlator is an optical computer for comparing two signals by utilising the Fourier transforming properties of a lens. It is commonly used in optics for target tracking and identification.
Analytic signal
In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued func
Modulated complex lapped transform
The modulated complex lapped transform (MCLT) is a lapped transform, similar to the modified discrete cosine transform, that explicitly represents the phase (complex values) of the signal.
In mathematical physics and harmonic analysis, the quadratic Fourier transform is an integral transform that generalizes the fractional Fourier transform, which in turn generalizes the Fourier transfo
Sparse Fourier transform
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, spectrum sensing and analog-to-digital
Fourier-transform infrared spectroscopy
Fourier-transform infrared spectroscopy (FTIR) is a technique used to obtain an infrared spectrum of absorption or emission of a solid, liquid, or gas. An FTIR spectrometer simultaneously collects hig
Algebraic analysis
Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of func
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra over the real or complex numbers (or over a non-Archimedean complete ) that at th
Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transfo
Finite Fourier transform
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., (pp. 52–53) describes the finite Fourier tra
Overlap–save method
In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal and a finite impulse response (FIR) filter : where h[m]
Fractional wavelet transform
Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform (WT). This transform is proposed in order to rectify the limitations of the WT and the fractional Fourier tra
Unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the
Spectral density
The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a num
List of window functions
In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several wi
Discrete cosine transform
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972,
Discrete sine transform
In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary part
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function that expresses how the shape of one is modified b
Compact group
In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within t
Planar Fourier capture array
A planar Fourier capture array (PFCA) is a tiny camera that requires no mirror, lens, focal length, or moving parts. It is composed of angle-sensitive pixels, which can be manufactured in unmodified C
Fourier transform on finite groups
In mathematics, the Fourier transform on finite groups is a generalization of the discrete Fourier transform from cyclic to arbitrary finite groups.
Bilinear time–frequency distribution
Bilinear time–frequency distributions, or quadratic time–frequency distributions, arise in a sub-field of signal analysis and signal processing called time–frequency signal processing, and, in the sta
Multidimensional transform
In mathematical analysis and applications, multidimensional transforms are used to analyze the frequency content of signals in a domain of two or more dimensions.
Linear canonical transformation
In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-di
Multiplier (Fourier analysis)
In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply
Cagniard–De Hoop method
In the mathematical modeling of seismic waves, the Cagniard–De Hoop method is a sophisticated mathematical tool for solving a large class of wave and diffusive problems in horizontally layered media.
Shapiro polynomials
In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums. In signal pro
Discrete Hartley transform
A discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discrete Fourier transform (DFT), with analogous applications in signal processing and relat
Marcinkiewicz interpolation theorem
In mathematics, the Marcinkiewicz interpolation theorem, discovered by Józef Marcinkiewicz, is a result bounding the norms of non-linear operators acting on Lp spaces. Marcinkiewicz' theorem is simila
Bragg plane
In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, , at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peak
Instantaneous phase and frequency
Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions. The instantaneous phase (also know
Van der Corput lemma (harmonic analysis)
In mathematics, in the field of harmonic analysis,the van der Corput lemma is an estimate for oscillatory integralsnamed after the Dutch mathematician J. G. van der Corput. The following result is sta
Topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for
List of Fourier analysis topics
This is a list of Fourier analysis topics. See also the list of Fourier-related transforms, and the list of harmonic analysis topics.
Graph Fourier transform
In mathematics, the graph Fourier transform is a mathematical transform which eigendecomposes the Laplacian matrix of a graph into eigenvalues and eigenvectors. Analogously to the classical Fourier Tr
Bicoherence
In mathematics and statistical analysis, bicoherence (also known as bispectral coherency) is a squared normalised version of the bispectrum. The bicoherence takes values bounded between 0 and 1, which
List of Fourier-related transforms
This is a list of linear transformations of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are sinuso
SAMV (algorithm)
SAMV (iterative sparse asymptotic minimum variance) is a parameter-free superresolution algorithm for the linear inverse problem in spectral estimation, direction-of-arrival (DOA) estimation and tomog
Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable (usually , in the time domain)
In mathematics and Fourier analysis, a rectangular mask short-time Fourier transform (rec-STFT) has the simple form of short-time Fourier transform. Other types of the STFT may require more computatio
Dirac delta function
In mathematics, the Dirac delta distribution (δ distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at
Fourier integral operator
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operat
Discrete Fourier series
In digital signal processing, the term Discrete Fourier series (DFS) is any periodic discrete-time signal comprising harmonically-related (i.e. Fourier) discrete real sinusoids or discrete complex exp
Solid harmonics
In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions . There are two kinds: the regular solid harmonic
Triple correlation
The triple correlation of an ordinary function on the real line is the integral of the product of that function with two independently shifted copies of itself: The Fourier transform of triple correla
Friedel's law
Friedel's law, named after Georges Friedel, is a property of Fourier transforms of real functions. Given a real function , its Fourier transform has the following properties. * where is the complex c
Discrete Fourier transform over a ring
In mathematics, the discrete Fourier transform over a ring generalizes the discrete Fourier transform (DFT), of a function whose values are commonly complex numbers, over an arbitrary ring.
Fourier operator
The Fourier operator is the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform, and is a two-dimensional function when it corresponds to the Fourier transf
Short-time Fourier transform
The short-time Fourier transform (STFT), is a Fourier-related transform used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. In practice,
Basis function
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as e
List of cycles
This is a list of recurring cycles. See also Index of wave articles, Time, and Pattern.
Fourier optics
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, or superposition, of plane waves. It has
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(nx) and cos(nx) with n taking on the values of
Laplace–Carson transform
In mathematics, the Laplace–Carson transform, named after Pierre Simon Laplace and John Renshaw Carson, is an integral transform with significant applications in the field of physics and engineering,