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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 measured by the spectral concentration. Spectral concentration The discrete Fourier transform (DFT) ''U''(''f'') of a finite series w_t, t = 1,2,3,4,...,T is defined as :U(f) = \sum_^w_t e^. In the following, the sampling interval will be taken as Δ''t'' = 1, and hence the frequency interval as ''f'' ∈ , ''U''(''f'') is a periodic function with a period 1. For a given frequency ''W'' such that 0\lambda_), then the eigenvector corresponding to \lambda_ is called ''nth''–order Slepian sequence (DPSS) (0≤''n''≤''N''-1). This ''nth''–order taper also offers the best sidelobe suppression and is pairwise orthogonal to the Slepian sequences of previous orders (0,1,2,3....,n-1). These lower order Slepian sequences form the basis for spectral estimation by multitaper method. N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
DPSS Figure
A diode-pumped solid-state laser (DPSSL) is a solid-state laser made by pumping a solid gain medium, for example, a ruby or a neodymium-doped YAG crystal, with a laser diode. DPSSLs have advantages in compactness and efficiency over other types, and high power DPSSLs have replaced ion lasers and flashlamp-pumped lasers in many scientific applications, and are now appearing commonly in green and other color laser pointers. Coupling The wavelength of laser diodes is tuned by means of temperature to produce an optimal compromise between the absorption coefficient in the crystal and energy efficiency (lowest possible pump photon energy). As waste energy is limited by the thermal lens this means higher power densities compared to high-intensity discharge lamps. High power lasers use a single crystal, but many laser diodes are arranged in strips (multiple diodes next to each other in one substrate) or stacks (stacks of substrates). This diode grid can be imaged onto the crystal b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No The Other Shannon Number
No or NO may refer to: Linguistics and symbols * ''Yes'' and ''no'', responses * No, an English determiner in noun phrases * No (kana) (, ), a letter/syllable in Japanese script * No symbol (🚫), the general prohibition sign * Numero sign ( or No.), a typographic symbol for the word "number" * Norwegian language (ISO 639-1 code "no") Places * Niederösterreich (''NÖ''), Lower Austria * Norway (ISO 3166-1 country code NO, internet top level domain .no) * No, Denmark, a village in Denmark * Nō, Niigata, a former town in Japan * No Creek (other), several streams * Lake No, in South Sudan * New Orleans, Louisiana, US or its professional sports teams: ** New Orleans Saints of the National Football League ** New Orleans Pelicans of the National Basketball Association * Province of Novara (Piedmonte, Italy), province code NO Arts and entertainment Film and television * ''No'' (2012 film), a 2012 Chilean film * ''Nô'' (film), a 1998 Canadian film * Julius No, the titular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the 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 into the intensities of its constituent 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. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', with the meaning of "a speaking of the world". In 1731, German philosopher Christian Wolff used the term cosmology in Latin (''cosmologia'') to denote a branch of metaphysics that deals with the general nature of the physical world. Religious or mythological cosmology is a body of beliefs based on mythological, religious, and esoteric literature and traditions of creation myths and eschatology. In the science of astronomy, cosmology is concerned with the study of the chronology of the universe. Physical cosmology is the study of the observable universe's origin, its large-scale structures and dynamics, and the ultimate fate of the universe, including the laws of science that govern these areas. It is investigated by scientists, including astronomers and physicists, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geophysics
Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct investigations across a wide range of scientific disciplines. The term ''geophysics'' classically refers to solid earth applications: Earth's figure of the Earth, shape; its gravitational, Earth's magnetic field, magnetic fields, and electromagnetic fields; its structure of the Earth, internal structure and Earth#Chemical composition, composition; its geodynamics, dynamics and their surface expression in plate tectonics, the generation of magmas, volcanism and rock formation. However, modern geophysics organizations and pure scientists use a broader definition that includes the water cycle including snow and ice; geophysical fluid dynamics, fluid dynamics of the oceans and the atmosphere; atmospheric electricity, electricity and magnetism in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 fields. The table of spherical harmonics contains a list of common spherical harmonics. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, every function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a cen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Estimation
In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities. Some SDE techniques assume that a signal is composed of a limited (usually small) number of generating frequencies plus noise and seek to find the location and intensity of the generated frequencies. Others make no assumption on the number of components and seek to estimate the whole generating spectrum. Overview Spectrum analysis, also referred to as frequency domain analysis or spectral density estimation, is the technical process of decomposing a complex signal into s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically used for lines and planes that intersect to form a right angle, whereas ''orthogonal'' is used in generalizations, such as ''orthogonal vectors'' or ''orthogonal curves''. ''Orthogonality'' is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics Optics In optics, polarization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive-definite Matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number \mathbf^\mathsf M \mathbf is positive for every nonzero real column vector \mathbf, where \mathbf^\mathsf is the row vector transpose of \mathbf. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \mathbf^* M \mathbf is positive for every nonzero complex column vector \mathbf, where \mathbf^* denotes the conjugate transpose of \mathbf. Positive semi-definite matrices are defined similarly, except that the scalars \mathbf^\mathsf M \mathbf and \mathbf^* M \mathbf are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''. Some authors use more general definitions of definiteness, permitting the matrices to be ... [...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] |
