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The generating functions are also known as scaling functions or father wavelets. In most cases one demands of those functions to be piecewise continuous with compact support .
A wavelet is a mathematical function used to divide a given function or continuous-time signal into different scale components. Usually one can assign a frequency range to each scale component. Each scale component can then be studied with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets.
The 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 ...
Both the scaling function (low-pass filter) and the wavelet function (high-pass filter) must be normalised by a factor /. Below are the coefficients for the scaling functions for C6–30. The wavelet coefficients are derived by reversing the order of the scaling function coefficients and then reversing the sign of every second one (i.e. C6 ...
Spectrum of the Meyer wavelet (numerically computed). The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer. [1] As a type of a continuous wavelet, it has been applied in a number of cases, such as in adaptive filters, [2] fractal random fields, [3] and multi-fault classification.
Scaling of the wavelet-basis-function by this factor and subsequent FFT of this function Multiplication with the transformed signal YFFT of the first step Inverse transformation of the product into the time domain results in Y W ( c , τ ) {\displaystyle Y_{W}(c,\tau )} for different discrete values of τ {\displaystyle \tau } and a discrete ...
Daubechies wavelets are widely used in solving a broad range of problems, e.g. self-similarity properties of a signal or fractal problems, signal discontinuities, etc. The Daubechies wavelets are not defined in terms of the resulting scaling and wavelet functions; in fact, they are not possible to write down in closed form.
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 using an iterative algorithm. It starts from values on a coarse sequence of sampling points and produces values for successively more densely spaced ...