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Legendre Wavelet
In functional analysis, compactly supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications in which spherical coordinate system is appropriate.Colomer and Colomer As with many wavelets there is no nice analytical formula for describing these harmonic spherical wavelets. The low-pass filter associated to Legendre multiresolution analysis is a finite impulse response (FIR) filter. Wavelets associated to FIR filters are commonly preferred in most applications. An extra appealing feature is that the Legendre filters are ''linear phase'' FIR (i.e. multiresolution analysis associated with linear phase filters). These wavelets have been implemented on MATLAB (wavelet toolbox). Although being compactly supported wavelet, legdN are not orthogonal (but for ''N'' = 1). Legendre multiresolution filters Associated Legendre polynomials are the colatitudinal part of the spherical harmonics w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word '' functional'' as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roll-off
Roll-off is the steepness of a transfer function with frequency, particularly in electrical network analysis, and most especially in connection with filter circuits in the transition between a passband and a stopband. It is most typically applied to the insertion loss of the network, but can, in principle, be applied to any relevant function of frequency, and any technology, not just electronics. It is usual to measure roll-off as a function of logarithmic frequency; consequently, the units of roll-off are either decibels per decade (dB/decade), where a decade is a tenfold increase in frequency, or decibels per octave (dB/8ve), where an octave is a twofold increase in frequency. The concept of roll-off stems from the fact that in many networks roll-off tends towards a constant gradient at frequencies well away from the cut-off point of the frequency curve. Roll-off enables the cut-off performance of such a filter network to be reduced to a single number. Note that roll-off ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet Packets
Originally known as optimal subband tree structuring (SB-TS), also called wavelet packet decomposition (WPD) (sometimes known as just wavelet packets or subband tree), is a wavelet transform where the discrete-time (sampled) signal is passed through more filters than the discrete wavelet transform (DWT). Introduction In the DWT, each level is calculated by passing only the previous wavelet approximation coefficients (''cAj'') through discrete-time low- and high-pass quadrature mirror filters. However, in the WPD, both the detail (''cDj'' (in the 1-D case), ''cHj'', ''cVj'', ''cDj'' (in the 2-D case)) and approximation coefficients are decomposed to create the full binary tree.Daubechies, I. (1992), Ten lectures on wavelets, SIAM. For ''n'' levels of decomposition the WPD produces 2''n'' different sets of coefficients (or nodes) as opposed to sets for the DWT. However, due to the downsampling process the overall number of coefficients is still the same and there is no redundanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figura Legd3
Figura may refer to: * Bella Figura, one act ballet by Jiří Kylián * Fgura, town in the south of Malta * Figura etymologica, rhetoric al figure * Figura Serpentinata, style in painting and sculpture * Oliva figura, species of sea snail, a marine gastropod mollusk in the family Olividae (olives) * translation of figure in some languages *Typology, a new testament theory of interpretation of events, people and sacraments of the Hebrew bible as figurative *''Figura,'' a 1938 essay by Erich Auerbach People * Anna Figura Anna Figura (born 6 February 1990) is a Polish ski mountaineer. Figura is born in Zakopane, and studies forestry at the University of Krakow.Monika StrojnyTrzy medale Polki na Mistrzostwach Europy . She is member of the ''Klub Skialpinistyczny K ... (b. 1990), Polish ski mountaineer * Katarzyna Figura (b. 1962), Polish actress * Paulina Figura (b. 1991), Polish ski mountaineer {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figura Legd2
Figura may refer to: * Bella Figura, one act ballet by Jiří Kylián * Fgura, town in the south of Malta * Figura etymologica, rhetoric al figure * Figura Serpentinata, style in painting and sculpture * Oliva figura, species of sea snail, a marine gastropod mollusk in the family Olividae (olives) * translation of figure in some languages *Typology, a new testament theory of interpretation of events, people and sacraments of the Hebrew bible as figurative *''Figura,'' a 1938 essay by Erich Auerbach People * Anna Figura Anna Figura (born 6 February 1990) is a Polish ski mountaineer. Figura is born in Zakopane, and studies forestry at the University of Krakow.Monika StrojnyTrzy medale Polki na Mistrzostwach Europy . She is member of the ''Klub Skialpinistyczny K ... (b. 1990), Polish ski mountaineer * Katarzyna Figura (b. 1962), Polish actress * Paulina Figura (b. 1991), Polish ski mountaineer {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Wavelet
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and extensively used as a teaching example. The Haar sequence was proposed in 1909 by Alfréd Haar. Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval , 1 The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1. The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property can, however, be an advantage for t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cascade Algorithm
In the mathematical topic of wavelet theory, the cascade algorithm is a numerical method for calculating function values of the basic scaling and wavelet functions of a discrete wavelet transform In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal ... using an iterative algorithm. It starts from values on a coarse sequence of sampling points and produces values for successively more densely spaced sequences of sampling points. Because it applies the same operation over and over to the output of the previous application, it is known as the ''cascade algorithm''. Successive approximation The iterative algorithm generates successive approximations to ψ(''t'') or φ(''t'') from and filter coefficients. If the algorithm converges to a fixed point, then that fixed point is the basic scali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrature Mirror Filter
In digital signal processing, a quadrature mirror filter is a filter whose magnitude response is the mirror image around \pi/2 of that of another filter. Together these filters, first introduced by Croisier et al., are known as the quadrature mirror filter pair. A filter H_1(z) is the quadrature mirror filter of H_0(z) if H_1(z) = H_0(-z). The filter responses are symmetric about \Omega = \pi / 2: : \big, H_1\big(e^\big)\big, = \big, H_0\big(e^\big)\big, . In audio/voice codecs, a quadrature mirror filter pair is often used to implement a filter bank that splits an input signal into two bands. The resulting high-pass and low-pass signals are often reduced by a factor of 2, giving a critically sampled two-channel representation of the original signal. The analysis filters are often related by the following formula in addition to quadrate mirror property: : \big, H_0\big(e^\big)\big, ^2 + \big, H_1\big(e^\big)\big, ^2 = 1, where \Omega is the frequency, and the sampling rate is nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre MRA Filter
Legendre, LeGendre or Le Gendre is a French surname. It may refer to: * Adrien-Marie Legendre (1752–1833), French mathematician ** Associated Legendre polynomials ** Legendre's equation ** Legendre polynomials ** Legendre symbol ** Legendre transformation ** Legendre (crater), a lunar impact crater located near the eastern limb of the Moon ** 26950 Legendre, a main-belt asteroid discovered on May 11, 1997 * Anne Legendre Armstrong (1927–2008), United States diplomat and politician * Charles Le Gendre (1830–1899), French-born American general and diplomat * François Legendre (1763–1853), surveyor, seigneur and political figure in Lower Canada * Géraldine Legendre (born 1953), French-American cognitive scientist and linguist * Gertrude Sanford Legendre (1902–2000), American socialite who served as a spy during World War II * Jacques Legendre (other), several people * Kevin Le Gendre, British journalist and broadcaster * Louis Legendre (1752–1797), French ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than 4 million users worldwide. They come from various backgrounds of engineering, science, and economics. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was based on his 1960s PhD thesis. Moler became a math professor at the University of New Mexico and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 different kinds of data, including but not limited to au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Phase
In signal processing, linear phase is a property of a filter where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time (usually delayed) by the same constant amount (the slope of the linear function), which is referred to as the group delay. Consequently, there is no phase distortion due to the time delay of frequencies relative to one another. For discrete-time signals, perfect linear phase is easily achieved with a finite impulse response (FIR) filter by having coefficients which are symmetric or anti-symmetric. Approximations can be achieved with infinite impulse response (IIR) designs, which are more computationally efficient. Several techniques are: * a Bessel transfer function which has a maximally flat group delay approximation function * a phase equalizer Definition A filter is called a linear phase filter if the phase component of the frequency response is a linear fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
