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The equation of a 1-D Gabor wavelet is a Gaussian modulated by a complex exponential, described as follows: [3] = / ()As opposed to other functions commonly used as bases in Fourier Transforms such as and , Gabor wavelets have the property that they are localized, meaning that as the distance from the center increases, the value of the function becomes exponentially suppressed.
The Morlet wavelet filtering process involves transforming the sensor's output signal into the frequency domain. By convolving the signal with the Morlet wavelet, which is a complex sinusoidal wave with a Gaussian envelope, the technique allows for the extraction of relevant frequency components from the signal.
The Gabor transform, named after Dennis Gabor, is a special case of the short-time Fourier transform.It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time.
In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. [1] [2] [3] [4]
There are far fewer significant components in the wavelet domain in this example than there are in the time domain, and most of the significant components are towards the coarser coefficients on the left. Hence, natural signals are compressible in the wavelet domain. The wavelet transform is a multiresolution, bandpass representation of a signal.
The wavelets forming a continuous wavelet transform (CWT) are subject to the uncertainty principle of Fourier analysis respective sampling theory: [4] given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event. The product of the uncertainties of time and frequency response ...
The fast wavelet transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain.
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. The graphs below are generated using the cascade algorithm, a numeric technique consisting of inverse-transforming [1 0 0 0 0 ... ] an appropriate number of times.