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In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. [1] The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a ...
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.
1D Gaussian wave packet, shown in the complex plane, for =, =, =, =. The overall group velocity is positive, and the wave packet moves as it disperses. The inverse Fourier transform is still a Gaussian, but now the parameter a has become complex, and there is an overall normalization factor.
There are three unknown parameters for a 1D Gaussian function (a, b, c) and five for a 2D Gaussian function (;,;,). The most common method for estimating the Gaussian parameters is to take the logarithm of the data and fit a parabola to the resulting data set.
The Gaussian function has a 1/e 2 diameter (2w as used in the text) about 1.7 times the FWHM.. At a position z along the beam (measured from the focus), the spot size parameter w is given by a hyperbolic relation: [1] = + (), where [1] = is called the Rayleigh range as further discussed below, and is the refractive index of the medium.
Full width at half maximum. In a distribution, full width at half maximum (FWHM) is the difference between the two values of the independent variable at which the dependent variable is equal to half of its maximum value.
The orbits' envelope of the projectiles (with constant initial speed) is a concave parabola. The initial speed is 10 m/s. We take g = 10 m/s 2. We consider the following example of envelope in motion. Suppose at initial height 0, one casts a projectile into the air with constant initial velocity v but different elevation angles θ.
Its impulse response is defined by a sinusoidal wave (a plane wave for 2D Gabor filters) multiplied by a Gaussian function. [6] Because of the multiplication-convolution property (Convolution theorem), the Fourier transform of a Gabor filter's impulse response is the convolution of the Fourier transform of the harmonic function (sinusoidal function) and the Fourier transform of the Gaussian ...