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Fourier Cosine Transform
Fourier may refer to: * Fourier (surname), French surname Mathematics *Fourier series, a weighted sum of sinusoids having a common period, the result of Fourier analysis of a periodic function * Fourier analysis, the description of functions as sums of sinusoids *Fourier transform, the type of linear canonical transform that is the generalization of the Fourier series *Fourier operator, the kernel of the Fredholm integral of the first kind that defines the continuous Fourier transform * Fourier inversion theorem, any one of several theorems by which Fourier inversion recovers a function from its Fourier transform * Short-time Fourier transform or short-term Fourier transform (STFT), a Fourier transform during a short term of time, used in the area of signal analysis *Fractional Fourier transform (FRFT), a linear transformation generalizing the Fourier transform, used in the area of harmonic analysis * Discrete-time Fourier transform (DTFT), the reverse of the Fourier series, a spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier (surname)
Fourier is a surname. Notable people with the surname include: * Charles Fourier (1772–1837), French utopian socialist thinker *Joseph Fourier Jean-Baptiste Joseph Fourier (; ; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre, Burgundy and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analys ... (1768–1830), French mathematician and physicist * Peter Fourier (1565–1640), French saint in the Roman Catholic Church and priest of Mattaincourt See also * Fourier (other) {{surname French-language surnames Occupational surnames Surnames of French origin Surnames of Belgian origin ... [...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] |
Fourier (crater)
Fourier is a lunar impact crater that is located in the southwestern part of the Moon's near side, just to the southeast of the crater Vieta. To the northeast is the Mare Humorum. The rim of this crater is roughly circular, but appears oval when viewed from the Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ... due to foreshortening. Except to the north-northwest, the outer rim is not heavily eroded. The satellite crater Fourier B lies along the eastern rim and inner side. The inner wall is relatively wide, and is slumped slightly along the edge forming a shelf near the perimeter. The interior floor is just over half the diameter of the crater rim, and is relatively level with a small crater to the west of the midpoint and another near the northeast edge. Satellite crater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Fourier-related Transforms
This is a list of linear transformations of function (mathematics), functions related to Fourier analysis. Such transformations Map (mathematics), map a function to a set of coefficients of basis functions, where the basis functions are trigonometric function, sinusoidal and are therefore strongly localized in the frequency spectrum. (These transforms are generally designed to be invertible.) In the case of the Fourier transform, each basis function corresponds to a single frequency component. Continuous transforms Applied to functions of continuous arguments, Fourier-related transforms include: * Two-sided Laplace transform * Mellin transform, another closely related integral transform * Laplace transform: the Fourier transform may be considered a special case of Two-sided Laplace transform#Relationship to the Fourier transform, the imaginary axis of the bilateral Laplace transform * Fourier transform, with special cases: ** Fourier series *** When the input function/waveform i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier-transform Spectroscopy
Fourier-transform spectroscopy (FTS) is a measurement technique whereby Spectrum (physics), spectra are collected based on measurements of the coherence (physics), coherence of a Radiation, radiative source, using time-domain or space-domain measurements of the radiation, electromagnetic radiation, electromagnetic or not. It can be applied to a variety of types of ''spectroscopy'' including optical spectroscopy, infrared spectroscopy (Fourier-transform infrared spectroscopy, FTIR, FT-NIRS), Nuclear Magnetic Resonance Spectroscopy, nuclear magnetic resonance (NMR) and magnetic resonance spectroscopic imaging (MRSI), mass spectrometry and electron spin resonance spectroscopy. There are several methods for measuring the temporal coherence of the light (see: Optical autocorrelation#Field autocorrelation, field-autocorrelation), including the continuous-wave and the pulsed Fourier-transform spectrometer or Fourier-transform spectrograph. The term "Fourier-transform spectroscopy" refl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Number
In the study of heat conduction, the Fourier number, is the ratio of time, t , to a characteristic time scale for heat diffusion, t_d . This dimensionless group is named in honor of J.B.J. Fourier, who formulated the modern understanding of heat conduction. The time scale for diffusion characterizes the time needed for heat to diffuse over a distance, L . For a medium with thermal diffusivity, \alpha , this time scale is t_d = L^2/\alpha , so that the Fourier number is t/t_d = \alpha t/L^2 . The Fourier number is often denoted as \mathrm or \mathrm_L . The Fourier number can also be used in the study of mass diffusion, in which the thermal diffusivity is replaced by the mass diffusivity. The Fourier number is used in analysis of time-dependent transport phenomena, generally in conjunction with the Biot number if convection is present. The Fourier number arises naturally in nondimensionalization of the heat equation. Definition The general definition of the Four ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Fourier Series
A generalized Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses an orthonormal basis of trigonometric functions, and the series expansion is applied to periodic functions. In contrast, a generalized Fourier series uses any set of orthogonal basis functions and can apply to any square integrable function. Definition Consider a set \Phi = \_^\infty of square-integrable complex valued functions defined on the closed interval ,b that are pairwise orthogonal under the weighted inner product: \langle f, g \rangle_w = \int_a^b f(x) \overline w(x) dx, where w(x) is a weight function and \overline g is the complex conjugate of g . Then, the generalized Fourier series of a function f is: f(x) = \sum_^\infty c_n\phi_n(x),where the coefficients are given by: c_n = . Sturm-Liouville Problems Given the space L^2(a,b) of square integrable functions defined on a given inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Fourier Transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). A Fourier transform converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by Matrix decomposition, factorizing the DFT matrix into a product of Sparse matrix, sparse (mostly zero) factors. As a result, it manages to reduce the Computational complexity theory, complexity of computing the DFT from O(n^2), which arises if one simply applies the definition of DFT, to O(n \log n), where is the data size. The difference in speed can be enormous, especially for long data sets where may be in the thousands or millions. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 discrete values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the fact that the transform operates on discrete data, often samples whose interval has units of time. From uniformly spaced samples it produces a function of frequency that is a periodic summation of the continuous Fourier transform of the original continuous function. In simpler terms, when you take the DTFT of regularly-spaced samples of a continuous signal, you get repeating (and possibly overlapping) copies of the signal's frequency spectrum, spaced at intervals corresponding to the sampling frequency. Under certain theoretical conditions, described by the sampling theorem, the original continuous function can be recovered perfectly from the DTFT and thus from the original discrete samples. The DTFT itself is a continuous functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 transform to the ''n''-th power, where ''n'' need not be an integer — thus, it can transform a function to any ''intermediate'' domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Wiener on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups. Since th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, the procedure for computing STFTs is to divide a longer time signal into shorter segments of equal length and then compute the Fourier transform separately on each shorter segment. This reveals the Fourier spectrum on each shorter segment. One then usually plots the changing spectra as a function of time, known as a spectrogram or waterfall plot, such as commonly used in software defined radio (SDR) based spectrum displays. Full bandwidth displays covering the whole range of an SDR commonly use fast Fourier transforms (FFTs) with 2^24 points on desktop computers. Forward STFT Continuous-time STFT Simply, in the continuous-time case, the function to be transformed is multiplied by a window function which is nonzero for only a short period of time. The Fourier transform (a o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
