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Fractional Wavelet Transform
Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform (WT). This transform is proposed in order to rectify the limitations of the WT and the fractional Fourier transform (FRFT). The FRWT inherits the advantages of multiresolution analysis A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introd ... of the WT and has the capability of signal representations in the fractional domain which is similar to the FRFT. Definition The fractional Fourier transform (FRFT), a generalization of the Fourier transform (FT), serves a useful and powerful analyzing tool in optics, communications, signal and image processing, etc. This transform, however, has one major drawback due to using global kernel, i.e., the fractional Fourier representation only provides such FRFT spect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet Transform
In mathematics, a wavelet series is a representation of a square-integrable (real number, real- or complex number, complex-valued) function (mathematics), function by a certain orthonormal series (mathematics), series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. Definition A function \psi \,\in\, L^2(\mathbb) is called an orthonormal wavelet if it can be used to define a Hilbert space#Orthonormal bases, Hilbert basis, that is a orthonormal basis, complete orthonormal system, for the Hilbert space L^2\left(\mathbb\right) of Square-integrable function, square integrable functions. The Hilbert basis is constructed as the family of functions \ by means of Dyadic transformation, dyadic translation (geometry), translations and dilation (operator theory), dilations of \psi\,, :\psi_(x) = 2^\frac \psi\left(2^jx - k\right)\, for integers j,\, k \,\in\, \mathbb. If under the standard ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 then, there h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiresolution Analysis
A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ''ironing method'') and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson anJames L. Crowley Definition A multiresolution analysis of the Lebesgue space L^2(\mathbb) consists of a sequence of nested subspaces ::\\dots\subset V_1\subset V_0\subset V_\subset\dots\subset V_\subset V_\subset\dots\subset L^2(\R) that satisfies certain self-similarity relations in time-space and scale-frequency, as well as completeness and regularity relations. * ''Self-similarity'' in ''time'' demands that each subspace ''Vk'' is invariant under sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelets
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 aud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
