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Fourier Operator
The Fourier operator is the integral kernel, kernel of the Fredholm integral equation#Equation of the first kind, Fredholm integral of the first kind that defines the continuous Fourier transform, and is a two-dimensional function when it corresponds to the Fourier transform of one-dimensional functions. It is complex-valued and has a constant (typically unity) magnitude everywhere. When depicted, e.g. for teaching purposes, it may be visualized by its separate real and imaginary parts, or as a colour image using a colour wheel to denote phase. It is usually denoted by a capital letter "F" in script font (\mathcal), e.g. the Fourier transform of a function g(t) would be written using the operator as \mathcalg(t). It may be thought of as a limiting case (mathematics), limiting case for when the size of the discrete Fourier transform increases without bound while its spatial resolution also increases without bound, so as to become both continuous and not necessarily periodic. Visua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Integral Kernel
In mathematics, an integral transform is a type of transform (mathematics), transform that maps a function (mathematics), function from its original function space into another function space via integral, integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any Transformation (function), transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function (mathematics), function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical Operator (mathematics), operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two Variable (mathematics), variables, that is called the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Fredholm Integral Equation
In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian. Equation of the first kind A Fredholm equation is an integral equation in which the term containing the kernel function (defined below) has constants as integration limits. A closely related form is the Volterra integral equation which has variable integral limits. An inhomogeneous Fredholm equation of the first kind is written as and the problem is, given the continuous kernel function K and the function g, to find the function f. An important case of these types of equation is the case when the kernel is a function only of the difference of its arguments, namely K(t,s)=K(t-s), and the limits of integration are ±∞, then the right hand side of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Limiting Case (mathematics)
In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example: * In statistics, the limiting case of the binomial distribution is the Poisson distribution. As the number of events tends to infinity in the binomial distribution, the random variable changes from the binomial to the Poisson distribution. *A circle is a limiting case of various other figures, including the Cartesian oval, the ellipse, the superellipse, and the Cassini oval. Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached. *Archimedes calculated an approximate value of π by treating the circle as the limiting case of a regular polygon with 3 × 2''n'' sides, as ''n'' gets large. *In electricity and magnetism, the long wavelength limit is the limiting case when the wave ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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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] [Amazon] |
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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] [Amazon] |
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Chirplet Transform
In signal processing, the chirplet transform is an inner product of an input signal with a family of analysis primitives called chirplets.S. Mann and S. Haykin,The Chirplet transform: A generalization of Gabor's logon transform, ''Proc. Vision Interface 1991'', 205–212 (3–7 June 1991).D. Mihovilovic and R. N. Bracewell, "Adaptive chirplet representation of signals in the time–frequency plane," ''Electronics Letters'' 27 (13), 1159–1161 (20 June 1991). Similar to the wavelet transform, chirplets are usually generated from (or can be expressed as being from) a single ''mother chirplet'' (analogous to the so-called '' mother wavelet'' of wavelet theory). Definitions The term ''chirplet transform'' was coined by Steve Mann, as the title of the first published paper on chirplets. The term ''chirplet'' itself (apart from chirplet transform) was also used by Steve Mann, Domingo Mihovilovic, and Ronald Bracewell to describe a windowed portion of a chirp function. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Least-squares Spectral Analysis
Least-squares spectral analysis (LSSA) is a method of estimating a Spectral density estimation#Overview, frequency spectrum based on a least-squares fit of Sine wave, sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in the long and gapped records; LSSA mitigates such problems. Unlike in Fourier analysis, data need not be equally spaced to use LSSA. Developed in 1969 and 1971, LSSA is also known as the Vaníček method and the Gauss-Vaniček method after Petr Vaníček, and as the Lomb method or the Lomb–Scargle periodogram, based on the simplifications first by Nicholas R. Lomb and then by Jeffrey D. Scargle. Historical background The close connections between Fourier analysis, the periodogram, and the least-squares fitting of sinusoids have been known for a long time. However, most developments are restricted to complete data sets of equally spaced samples. In 1963, Free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Integral Transforms
In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the ''inverse transform''. General form An integral transform is any transform ''T'' of the following form: :(Tf)(u) = \int_^ f(t)\, K(t, u)\, dt The input of this transform is a function ''f'', and the output is another function ''Tf''. An integral transform is a particular kind of mathematical operator. There are numerous useful integral transforms. Each is specified by a choice of the function K of two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated ''inverse kernel'' K^( u,t ) which (roughly speaking) yields an inverse transform: : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |