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For the case of input process $\{X(t)\}$ being white Gaussian noise with two-sided power spectral density $\frac{N_0}{2}$, the output process $\{Y(t)\}$ is a strictly stationary zero-mean Gaussian process in which all the random variables have the same variance $\frac{N_0}{2}\int_{-\infty}^\infty |H(f)|^2 \,\mathrm df$ almost as you say. But the key point ...


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so MJ, my approach to using the Gaussian window for the STFT is first to use the "ordinary frequency" definition of the continuous-time Fourier Transform: $$ \mathscr{F}\Big\{x(t)\Big\} \triangleq X(f) = \int\limits_{-\infty}^{\infty}x(t)\, e^{-j 2 \pi f t} \,\mathrm{d}t$$ $$ \mathscr{F}^{-1}\Big\{X(f)\Big\} \triangleq x(t) = \int\limits_{-\infty}^{\infty}...


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Why Does 2D FFT of Gaussian Looks More Sharper than Gaussian Itself? Have a look at the Fourier Transfrom of a Gaussian Signal. $$ \mathcal{F}_{x} \left\{ {e}^{-a {x}^{2} } \right\} \left( \omega \right) = \sqrt{\frac{\pi}{a}} {e}^{- {\pi}^{2} \frac{ {\omega}^{2} }{a} } $$ First, Gaussian Signal stays Gaussian under Fourier Transform. As you can see, the ...


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Questioner's answer... sigma have the same units as x and y i.e. number of pixels. In multi-scale filtering, the size of the filter must change when the sigma changes. Obtain the number of pixels per one millimeter or the vice-versa. (I did this using the property of pixel spacing included in the DICOM metadata in Matlab you can do this as info=dicominfo('...


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The binomial theorem is not necessarily involved: the top waveform is simply pointwise raised to the 5-th power, as others have noted, and there are no missing peaks that should be seen. To illustrate, consider the following crude hand-drawn figure: I drew this in my simulation software, so it was automatically "digitized" as I drew it. Obviously, I am poor ...


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Histograms of images can differ, widely. However, when features are inspected, one often uses derivative filters at different scales, or morphological decompositions, or independent component analysis. A traditional and heuristic model for the resulting coefficients of a component is that of the Generalized Gaussian-Laplacian Distribution, or GGD: $$ C_{...


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Okay, I am not quite understanding your question. Let's start with the definition of the Gaussian, aka the Bell Curve, in its general form. $$ f(t) = \frac{1}{ \sigma \sqrt{2\pi}} e^{ -\frac{(t-\mu)^2}{2\sigma^2} } $$ $\mu$ is the mean, and represents where the peak occurs. $\sigma$ is the standard deviation, and identifies where the inflection points ...


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then according to the binomial theorem, we should get 6 peaks Where does that statement come from and why do you think it applies here? Applying the binomial theorem we get terms that consists of Gaussian raised to different powers. However, when raising the power on a Gaussian, the peak stays where it is so there is no mechanism that would move it ...


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Tables 7.1 and 7.2 from Lattice Coding of Signals and Networks by Zamir are essentially this. Also the figures in this paper: https://arxiv.org/pdf/1103.0171.pdf


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I think that the question by OP xvan has several misconceptions in it and these propagate into the answer that he has provided for his own question. Define \begin{align}\phi_1(t) &= \sqrt{\frac 2T}\cos(2\pi f_ct),~ 0 \leq t \leq T,\\ \phi_2(t) &= -\sqrt{\frac 2T}\sin(2\pi f_ct),~ 0 \leq t \leq T, \end{align} where $f_c$ is the carrier frequency are ...


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okay, i am gonna change the variables a little and the names of variables, to make this more consistent with bone-head electrical engineers (which is me). my signal is $$ x(t) = e^{j \alpha t^2} $$ and my window is $$ w(t-u) = \left( \pi\sigma^2 \right)^{-1/4} \ e^{\frac{-(t-u)^2}{2\sigma^2}} $$ the product is $$\begin{align} x(t)w(t-u) &= e^{j \...


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You can easily have a look on the values of the STD on Image Denoising Papers: The range of 1-15 is considered low. The range 15-30 is considered medium. The range 30-50 (Even above) is considered high. The above values is for images in the range [0, 255].


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Have a look at my Fast Gaussian Blur Project at GitHub. You will find there implementation of IIR Approximation of Gaussian Blur which implements the following papaers: Recursive Gabor Filtering. Recursive Implementation of the Gaussian Filter. Boundary Conditions for Young - van Vliet Recursive Filtering. The idea is pretty straight forward.


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Have a look at https://doi.org/10.1063/1.3504369 In this reference the authors check for gaussian behavior of the 1/f noise of a resistor using a fourth order frequency spectrum.


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