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Wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of middle C and a short duration of roughly one tenth of a second. If this wavelet were to be convolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. Correlation is at the core of many practical wavelet applications. As a mathematical tool, wavelets can be used to extract information from many kinds of data, including audio signals and images. Sets of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alex Grossmann
Alexander Grossmann (5 August 1930 – 12 February 2019) was a French-American physicist of Croatian origin. Early life Aleksandar Grossmann was born to a Jewish family in Zagreb, where he was attending a gymnasium when World War II in Yugoslavia started. Forced to leave school and convert to Catholicism under the rule of the so-called Independent State of Croatia, he and his family fled to Montecatini Terme in Italy, and then Geneva in Switzerland. After the war, he returned to Zagreb and completed high school as well as graduated mathematics at the Faculty of Science, University of Zagreb in 1952. Career Grossmann started work at the Ruđer Bošković Institute, and collaborated with international visiting scholars between 1952 and 1955. He travelled to the United States in 1955, working in the physics departments of the Institute for Advanced Study (IAS), Princeton, Brandeis University, and the Courant Institute, NYU, then again at the IAS until 1963. After one year at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent States In Mathematical Physics
Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see alsoJ-P. Gazeau,''Coherent States in Quantum Physics'', Wiley-VCH, Berlin, 2009.). However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys.S.T. Ali, J-P. Antoine, J-P. Gazeau, and U.A. Mueller, Coherent states and their generalizations: A mathematical overview, ''Reviews in Mathematical Physics'' 7 (1995) 1013-1104.S.T. Ali, J-P. Antoine, and J-P. Gazeau, ''Coherent States, Wavelets and Their Generalizations'', Springer-Verlag, New York, Berlin, Heidelberg, 2000. A general definition Let \mathfrak H\, be a complex, separable Hilbert space, X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term ''convolution'' refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f*g differs from cross-correlation f \star g only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus i ... [...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 Alexander Grossmann (5 August 1930 – 12 February 2019) was a French-American physicist of Croatian origin. Early life Aleksandar Grossmann was born to a Jewish family in Zagreb, where he was attending a gymnasium when World War II in Yugoslav ... to develop what is now known as the Wavelet transform. Biography Morlet studied at from 1952 to 1955 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 Math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave
In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a travelling wave; by contrast, a pair of superposition principle, superimposed periodic waves traveling in opposite directions makes a ''standing wave''. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero. There are two types of waves that are most commonly studied in classical physics: mechanical waves and electromagnetic waves. In a mechanical wave, Stress (mechanics), stress and Strain (mechanics), strain fields oscillate about a mechanical equilibrium. A mechanical wave is a local deformation (physics), deformation (strain) in some physical medium that propa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interference (wave Propagation)
In physics, interference is a phenomenon in which two coherent waves are combined by adding their intensities or displacements with due consideration for their phase difference. The resultant wave may have greater amplitude (constructive interference) or lower amplitude (destructive interference) if the two waves are in phase or out of phase, respectively. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves as well as in loudspeakers as electrical waves. Etymology The word ''interference'' is derived from the Latin words ''inter'' which means "between" and ''fere'' which means "hit or strike", and was used in the context of wave superposition by Thomas Young in 1801. Mechanisms The principle of superposition of waves states that when two or more propagating waves of the same type are incident on the same point, the resultant amplitude at that point is equal to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavefront
In physics, the wavefront of a time-varying ''wave field (physics), field'' is the set (locus (mathematics), locus) of all point (geometry), points having the same ''phase (waves), phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal frequency (otherwise the phase is not well defined). Wavefronts usually move with time. For waves propagating in a dimension (mathematics), unidimensional medium, the wavefronts are usually single points; they are curves in a two dimensional medium, and surface (mathematics), surfaces in a three-dimensional one. For a sinusoidal plane wave, the wavefronts are planes perpendicular to the direction of propagation, that move in that direction together with the wave. For a sinusoidal spherical wave, the wavefronts are spherical surfaces that expand with it. If the speed of propagation is different at different points of a wavefront, the shape and/or orientation of the wavefro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherence (physics)
Coherence expresses the potential for two waves to Wave interference, interfere. Two Monochromatic radiation, monochromatic beams from a single source always interfere. Wave sources are not strictly monochromatic: they may be ''partly coherent''. When interfering, two waves add together to create a wave of greater amplitude than either one (constructive Wave interference, interference) or subtract from each other to create a wave of minima which may be zero (destructive interference), depending on their relative phase (waves), phase. Constructive or destructive interference are limit cases, and two waves always interfere, even if the result of the addition is complicated or not remarkable. Two waves with constant relative phase will be coherent. The amount of coherence can readily be measured by the interference visibility, which looks at the size of the interference fringes relative to the input waves (as the phase offset is varied); a precise mathematical definition of the de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelength
In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves), phase'' on the wave, such as two adjacent crests, troughs, or zero crossings. Wavelength is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The multiplicative inverse, inverse of the wavelength is called the ''spatial frequency''. Wavelength is commonly designated by the Greek letter lambda (''λ''). For a modulated wave, ''wavelength'' may refer to the carrier wavelength of the signal. The term ''wavelength'' may also apply to the repeating envelope (mathematics), envelope of modulated waves or waves formed by Interference (wave propagation), interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed phase velocity, wave speed, wavelength is inversely proportion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
French Language
French ( or ) is a Romance languages, Romance language of the Indo-European languages, Indo-European family. Like all other Romance languages, it descended from the Vulgar Latin of the Roman Empire. French evolved from Northern Old Gallo-Romance, a descendant of the Latin spoken in Northern Gaul. Its closest relatives are the other langues d'oïl—languages historically spoken in northern France and in southern Belgium, which French (Francien language, Francien) largely supplanted. It was also substratum (linguistics), influenced by native Celtic languages of Northern Roman Gaul and by the Germanic languages, Germanic Frankish language of the post-Roman Franks, Frankish invaders. As a result of French and Belgian colonialism from the 16th century onward, it was introduced to new territories in the Americas, Africa, and Asia, and numerous French-based creole languages, most notably Haitian Creole, were established. A French-speaking person or nation may be referred to as Fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
