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Morlet Wavelet
In mathematics, the Morlet wavelet (or Gabor wavelet)0). The parameter \sigma in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction \sigma>5 is used to avoid problems with the Morlet wavelet at low \sigma (high temporal resolution). For signals containing only slowly varying frequency and amplitude modulations (audio, for example) it is not necessary to use small values of \sigma. In this case, \kappa_ becomes very small (e.g. \sigma>5 \quad \Rightarrow \quad \kappa_5, the frequency of the Morlet wavelet is conventionally taken to be \omega_\simeq\sigma. The wavelet exists as a complex version or a purely real-valued version. Some distinguish between the "real Morlet" vs the "complex Morlet". Others consider the complex version to be the "Gabor wavelet", while the real-valued version is the "Morlet wavelet". Uses Use in medicine In magnetic resonance spectroscopy imaging, the Morlet wavelet transform method offers an in ... [...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] |
Constant-Q Transform
In mathematics and signal processing, the 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 transformJudith C. BrownCalculation of a constant Q spectral transform ''J. Acoust. Soc. Am.'', 89(1):425–434, 1991. and very closely related to the complex Morlet wavelet transform. Its design is suited for musical representation. The transform can be thought of as a series of filters ''f''''k'', logarithmically spaced in frequency, with the ''k''-th filter having a spectral width ''δf''''k'' equal to a multiple of the previous filter's width: :\delta f_k = 2^ \cdot \delta f_ = \left( 2^ \right)^k \cdot \delta f_\text, where ''δf''''k'' is the bandwidth of the ''k''-th filter, ''f''min is the central frequency of the lowest filter, and ''n'' is the number of filters per octave. Calculation The short-time Fourier transform of ''x'' 'n''for a frame shifted to sample ''m'' is cal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pitch Estimation
Pitch may refer to: Acoustic frequency * Pitch (music), the perceived frequency of sound including "definite pitch" and "indefinite pitch" ** Absolute pitch or "perfect pitch" ** Pitch class, a set of all pitches that are a whole number of octaves apart ** Relative pitch, the ability to identify a given musical interval between two notes * Pitch accent, a form of accentuation in speech Business * Sales pitch, a line of talk that attempts to persuade someone or something ** Pitch (filmmaking), a proposal for a film ** Elevator pitch, a very short sales presentation, allegedly short enough to be made during an elevator ride Measurement Movement about the transverse axis * Pitch angle (or pitch rotation), one of the angular degrees of freedom of any stiff body (for example a vehicle), describing rotation about the side-to-side axis ** Pitch (aviation), one of the aircraft principal axes of rotation (nose-up or nose-down angle measured from horizontal axis) ** Pitch (ship motion), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Induction Decay
In Fourier transform nuclear magnetic resonance spectroscopy, free induction decay (FID) is the observable NMR signal generated by non-equilibrium nuclear spin magnetization precessing about the magnetic field (conventionally along z). This non-equilibrium magnetization can be created generally by applying a pulse of radio-frequency close to the Larmor frequency of the nuclear spins. If the magnetization vector has a non-zero component in the xy plane, then the precessing magnetisation will induce a corresponding oscillating voltage in a detection coil surrounding the sample. This time-domain signal (a sinusoid) is typically digitised and then Fourier transformed in order to obtain a frequency spectrum of the NMR signal i.e. the NMR spectrum. The duration of the NMR signal is ultimately limited by T2 relaxation, but mutual interference of the different NMR frequencies present also causes the signal to be damped more quickly. When NMR frequencies are well-resolved, as is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed-point Iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed-point iteration is :x_=f(x_n), \, n=0, 1, 2, \dots which gives rise to the sequence x_0, x_1, x_2, \dots of iterated function applications x_0, f(x_0), f(f(x_0)), \dots which is hoped to converge to a point x_. If f is continuous, then one can prove that the obtained x_ is a fixed point of f, i.e., :f(x_)=x_ . More generally, the function f can be defined on any metric space with values in that same space. Examples * A first simple and useful example is the Babylonian method for computing the square root of ''a''>0, which consists in taking f(x)=\frac 12\left(\frac ax + x\right), i.e. the mean value of ''x'' and ''a/x'', to approach the limit x = \sqrt a (from whatever starting point x_0 \gg 0 ). This is a special case of Newton's method quoted b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually tapering away from the middle. Mathematically, when another function or waveform/data-sequence is "multiplied" by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the "view through the window". Equivalently, and in actual practice, the segment of data within the window is first isolated, and then only that data is multiplied by the window function values. Thus, tapering, not segmentation, is the main purpose of window functions. The reasons for examining segments of a longer function include detection of transient events and time-averaging of frequency spectra. The duration of the segments is determined in ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous adjective ''Gaussian'' is pronounced . Mathematics Algebra and linear algebra Geometry and differential geometry Number theory Cyclotomic fields *Gaussian period *Gaussian rational *Gauss sum, an exponential sum over Dirichlet characters ** Elliptic Gauss sum, an analog of a Gauss sum **Quadratic Gauss sum Analysis, numerical analysis, vector calculus and calculus of variations Complex analysis and convex analysis *Gauss–Lucas theorem *Gauss's continued fraction, an analytic continued fraction derived from the hypergeometric functions * Gauss's criterion – described oEncyclopedia of Mathematics* Gauss's hypergeometric theorem, an identity on hypergeometric series *Gauss plane Statistics *Gauss–Kuzmi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Wavelet Transform
Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous game, a generalization of games used in game theory ** Law of Continuity, a heuristic principle of Gottfried Leibniz * Continuous function, in particular: ** Continuity (topology), a generalization to functions between topological spaces ** Scott continuity, for functions between posets ** Continuity (set theory), for functions between ordinals ** Continuity (category theory), for functors ** Graph continuity, for payoff functions in game theory * Continuity theorem may refer to one of two results: ** Lévy's continuity theorem, on random variables ** Kolmogorov continuity theorem, on stochastic processes * In geometry: ** Parametric continuity, for parametrised curves ** Geometric continuity, a concept primarily applied to the conic secti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Morlet
Jean Morlet (; 13 January 1931 – 27 April 2007) was a French geophysicist who pioneered work in the field of wavelet analysis around the year 1975. He invented the term ''wavelet'' to describe the functions he was using. In 1981, Morlet worked with Alex Grossmann to develop what is now known as the Wavelet transform. Biography Morlet graduated from the École Polytechnique in 1952 and was research engineer at Elf Aquitaine when he invented wavelets to solve signal processing problems for oil prospecting. Awards He was awarded in 1997 with the Reginald Fessenden Award. He was awarded in 2001 with the first prize Prix Chéreau Lavet, from the Académie des Technologies. Legacy The Jean-Morlet Chair at the Centre International de Rencontres Mathématiques The Centre International de Rencontres Mathématiques (CIRM) is a mathematics research institute associated with the Centre National de la Recherche Scientifique (CNRS) and the Société Mathématique de France (SMF). It is lo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabor 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 and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window function, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis.E. Sejdić, I. Djurović, J. Jiang, “Time-frequency feature representation using energy concentration: An overview of recent advances,” ''Digital Signal Processing'', vol. 19, no. 1, pp. 153-183, January 2009. The window function means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal ''x''(''t'') is defined by this formula: : G_x(\tau,\omega) = \int_^\infty x(t)e^e^\,dt The Gaussian function has infinite range and it is impractical for implementation. However, a level of signific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
