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List Of Fourier Analysis Topics
{{Short description, none This is a list of Fourier analysis topics. Fourier analysis * Multiplier (Fourier analysis) * Fourier shell correlation * Pinsky phenomenon Fourier series * Generalized Fourier series * Regressive discrete Fourier series * Gibbs phenomenon * Sigma approximation * Dini test * Poisson summation formula * Spectrum continuation analysis * Convergence of Fourier series Fourier transforms * List of Fourier-related transforms * Fourier transform on finite groups * Fractional Fourier transform * Continuous Fourier transform * Fourier operator * Fourier inversion theorem * Sine and cosine transforms * Parseval's theorem * Paley–Wiener theorem * Projection-slice theorem * Frequency spectrum Discrete Fourier transforms * Discrete Fourier series * Non-uniform discrete Fourier transform * DFT matrix * Entropy influence conjecture * Fast cosine transform * Fast Fourier transform * Cyclotomic fast Fourier transform * Sparse Fourier transform Applications of Fouri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. 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. The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term ''Fourier an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function (mathematics), function as input then outputs another function that describes the extent to which various Frequency, frequencies are present in the original function. The output of the transform is a complex number, complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the Operation (mathematics), mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical Chord (music), chord into the sound intensity, intensities of its constituent Pitch (music), pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the #Uncertainty principle, uncerta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast 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, is a widely used transformation technique in signal processing and data compression. It is used in most digital media, including digital images (such as JPEG and HEIF), digital video (such as MPEG and ), digital audio (such as Dolby Digital, MP3 and AAC), digital television (such as SDTV, HDTV and VOD), digital radio (such as AAC+ and DAB+), and speech coding (such as AAC-LD, Siren and Opus). DCTs are also important to numerous other applications in science and engineering, such as digital signal processing, telecommunication devices, reducing network bandwidth usage, and spectral methods for the numerical solution of partial differential equations. A DCT is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using only real numbers. The DCTs are gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entropy Influence Conjecture
In mathematics, the entropy influence conjecture is a statement about Boolean functions originally conjectured by Ehud Friedgut and Gil Kalai in 1996. Statement For a function f: \^n \to \, note its Fourier expansion : f(x) = \sum_ \widehat(S) x_S, \text x_S = \prod_ x_i. The entropy–influence conjecture states that there exists an absolute constant ''C'' such that H(f) \leq C I(f), where the total influence I is defined by : I(f) = \sum_S , S, \widehat(S)^2, and the entropy H (of the spectrum) is defined by : H(f) = - \sum_S \widehat(S)^2 \log \widehat(S)^2 , (where ''x'' log ''x'' is taken to be 0 when ''x'' = 0). See also * Analysis of Boolean functions In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on \^n or \^n (such functions are sometimes known as pseudo-Boolean functions) from a spectral perspective. The functions studied a ... References Unsolved Problems i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DFT Matrix
In applied 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 through matrix multiplication. Definition An ''N''-point DFT is expressed as the multiplication X = W x, where x is the original input signal, W is the ''N''-by-''N'' square matrix, square DFT matrix, and X is the DFT of the signal. The square matrix ensures the transformation is invertable. The transformation matrix W can be defined as W = \left(\frac\right)_ , or equivalently: : W = \frac \begin 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^ \\ 1&\omega^2&\omega^4&\omega^6&\cdots&\omega^\\ 1&\omega^3&\omega^6&\omega^9&\cdots&\omega^\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 1&\omega^&\omega^&\omega^&\cdots&\omega^ \end , where \omega = e^ is a Root of unity, primitive ''N''th root of unity in which i^2=-1. We can avoid writing large exponents for \omega using the fact that for any exp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-uniform Discrete Fourier Transform
In applied mathematics, the non-uniform 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, but in which the input signal is not sampled at equally spaced points or frequencies (or both). It is a generalization of the shifted DFT. It has important applications in signal processing, magnetic resonance imaging, and the numerical solution of partial differential equations. As a generalized approach for nonuniform sampling, the NUDFT allows one to obtain frequency domain information of a finite length signal at any frequency. One of the reasons to adopt the NUDFT is that many signals have their energy distributed nonuniformly in the frequency domain. Therefore, a nonuniform sampling scheme could be more convenient and useful in many digital signal processing applications. For example, the NUDFT provides a variable spectral resolution controlled by the user. Definition The '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Series
In digital signal processing, a discrete Fourier series (DFS) is a Fourier series whose sinusoidal components are functions of a discrete variable instead of a continuous variable. The result of the series is also a function of the discrete variable, i.e. a discrete sequence. A Fourier series, by nature, has a discrete set of components with a discrete set of coefficients, also a discrete sequence. So a DFS is a representation of one sequence in terms of another sequence. Well known examples are the Discrete Fourier transform In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discre ... and its inverse transform. Introduction Relation to Fourier series The exponential form of Fourier series is given by: :s(t) = \sum_^\infty S cdot e^, which is periodic with an arbitrary period denoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Fourier Transform
In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced Sampling (signal processing), samples of a function (mathematics), function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex number, complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex number, complex Sine wave, sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Spectrum
In signal processing, the power spectrum S_(f) of a continuous time signal x(t) describes the distribution of power into frequency components f composing that signal. 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. The statistical average of any sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum. When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the energy spectral density. More commonly used is the power spectral density (PSD, or simply power spectrum), which applies to signals existing over ''all'' time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The PSD then refers to the spectral energy distribution that would b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection-slice Theorem
In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: * Take a two-dimensional function ''f''(r), Projection (mathematics), project (e.g. using the Radon 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 transform first, and then slice it through its origin, which is parallel to the projection line. In operator terms, if * ''F''1 and ''F''2 are the 1- and 2-dimensional Fourier transform operators mentioned above, * ''P''1 is the projection operator (which projects a 2-D function onto a 1-D line), * ''S''1 is a slice operator (which extracts a 1-D central slice from a function), then : F_1 P_1 = S_1 F_2. This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medical computed axial tomography, CT scans where a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paley–Wiener Theorem
In mathematics, a Paley–Wiener theorem is a theorem that relates decay properties 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) who, in 1934, introduced various versions of the theorem. The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration). The original work by Paley and Wiener is also used as a namesake in the fields of control theory and harmonic analysis; introducing the Paley–Wiener condition for spectral factorization and the Paley–Wiener criterion for non-harmonic Fourier series respectively. These are related mathematical concepts that place the decay properties of a function in context of stability pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parseval's Theorem
In 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 function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh. Although the term "Parseval's theorem" is often used to describe the unitarity of ''any'' Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem. Statement of Parseval's theorem Suppose that A(x) and B(x) are two complex-valued functions on \mathbb of period 2 \pi that are square integrable (with respect to the Lebesgue measure) over intervals of period length, with Fourier series :A(x)=\sum_^\infty a_ne^ and :B(x)=\sum_^\infty b_ne^ respecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
