Questions tagged [gaussian]
The gaussian function, an exponential function with a negative square of the argument in the exponent, is interesting in signal processing because the Fourier Transform of a gaussian function is also a gaussian function.
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Discrete Fourier Transform of the Gaussian
Cross-posted from here
I encountered the following question in a Digital Image Processing examination:
Find the 2D DFT of $\frac{1}{2 \pi \sigma^2} e^{-\frac{(x - x_0)^2 + (y - y_0)^2}{2 \sigma^2}}$ ...
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Finding mean, autocorrelation, and power spectrum of $y[n]=(x∗h)[n]$ where $x[n]$ is zero mean white gaussian noise
If given $x[n]$ is zero mean white Gaussian noise with variance $\sigma_x^2$, and a filter with a known impulse response $h[n]$, how would I find the mean, autocorrelation, and power spectrum of $y[n] ...
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What's the distribution of the DFT of a real-valued, zero-mean, normally distributed random vector?
Suppose $X$ is a real-valued N-dimensional Gaussian vector, $X \sim \mathcal{N}(\mathbf{0}, C_X)$. The discrete Fourier transform can be obtained by left-multiplying with the unitary DFT matrix, i.e. $...
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Approximating variable-length coding by "aggregating" fixed-length coding
Consider the problem where I want to quantize a univariate standard Gaussian with two bits, and the objective is to minimize MSE reconstruction loss. According to these slides, the Lloyd algorithm and ...
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Decomposing Gaussian Diffraction Pattern
I was recommended to post this question here from the physics stack exchange. I have the following diffraction pattern produced by a fabry-perot etalon.
The red represents the center of that pattern. ...
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Is it possible to reformulate a Kalman Filter as a Gaussian Markov Random Field?
The generic formulation of a KF uses a set of transitition equations, while the GMRF is typically specified through its mean and precision. However, a simple KF involves Gaussianity and Markov ...
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Generating time-domain noise from PSD
I have a question regarding generating a time-domain noise from the power spectral density (PSD), this was addressed in this question (How to generate time-series from a given one-sided PSD?) but I ...
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Can the standard deviation of the Gaussian window in a Gabor filter be made infinitesimally small?
My understanding is that the standard deviation of the Gaussian window in a Gabor filter dictates the temporal resolution.
Wouldn't it always be better then to make the window smaller, thus achieving ...
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Which probability density functions to use in a likelihood ratio test for the radar detection problem?
I am trying to understand the radar detection problem in the form of the generalized likelihood ratio test and am having a little trouble with understanding the noise distributions. Perhaps this will ...
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How to evaluate pairwise error probability and detection in presence of gaussian noise?
I am reading Digital Communication Systems by Simon Haykin and I am stuck at one point.
Consider a two-dimensional signal space that has a message constellation of four points, given by $s_1,s_2,s_3,...
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Spectral power density of thermal noise
In telecommunications the received signal $y(t)$ of a receiver is equal to $x(t)S+n(t)$ where $x(t)$ is the sent signal of the sender , $S$ is a constant which describes the total attenuation of the ...
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Approximating fractional-octave Gaussian smoothing with non-causal variable-width IIR filters
I am trying to implement fractional-octave smoothing of amplitude response data with approximated Gaussian filters, as briefly discussed in this AES paper.
Unfortunately, no implementation details are ...
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Contradiction when simulating WGN
To my knowledge, white Gaussian noise (WGN) is defined as a process with a correlation function:
$$
R[k]=\sigma^2 \delta[k]
$$
and whose symbols are distributed according to $N(0,\sigma^2)$.
Naturally,...
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Gaussian filter: Plotting DTFT and DFT (by hand) from the continuous-time impulsive response
I am trying to make an algorithm that plots out the Discrete-Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT) of the Gaussian filter. The impulsive response and its transfer ...
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Can a time signal be modeled as a multivariate Gaussian distribution?
Suppose I have several groups of signal measurements, each containing multiple replicates, and I know that within each group the signal "shape" is approximately the same but with variance/...
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Kalman Filter Under Non-Gaussian Noise
I know that Kalman filter is optimal filter under some assumption like process and measurement noise are Gaussian. But if the process and measurement noise is non-Gaussian, the estimation of the ...
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Real-time convolution with Gaussian noise
I have a brain activity simulator that is capable of receiving various stimuli. Both generated signals and input stimulus are causal: a single sample is created every time step. I use a sinusoidal ...
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Choosing sigma values for Gaussian blurring on an anisotropic image
I have an anisotropic image that is anisotropic both in terms of number of voxels and also in terms of voxel resolution.
...
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The Effect of the Finite Radius of Gaussian Kernel
Page 168 of Digital Image Processing, Global Edition says:
we know that the values of a Gaussian function at a
distance larger than 3𝜎 from the mean are small enough that they can be ignored.
If we ...
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Is f/fL a high pass filter where fL is a low-pass version of f?
Let there be a signal f and its low pass filtered signal be fL. Then what can we say about the spectrum of f/fL ?
To be specific I am obtaining fL simply using gaussian blurring in the spatial domain.
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Inverting the normalized Gaussian to get a kernel window radius
I am seeking to compute a kernel radius to use with my gaussian convolution filter, and inspired by https://stackoverflow.com/a/68050503/, I came up with:
$$r=\sqrt{-2\sigma^2\ln\left( \epsilon\sigma\...
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One-sided bandwidth of the Gaussian filter
As defined in the CCSDS (section 3.1.2), the impulse response of the Gaussian filter is given by
$$
h(t) = \frac{1}{\sigma T \sqrt{2\pi}}e^{-\frac{t^2}{2\sigma^2T^2}}
$$
where
$$
\sigma = \frac{\sqrt{\...
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Downsample (aggregate) raster by a non-integer factor, using a Gaussian filter kernel
The task is to downsample (aggregate) a raster from 100m pixel size to 460m. The aggregation should be performed using a Gaussian filter. To better understand the task, I am following the paper ‘The ...
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Finding the average probability of error for a transmission system using this signal constellation [duplicate]
We have a signal constellation with decision regions given in the figure below and symbols are equally likely to be transmitted.
Now we have to find the average probability of error for a ...
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ICA and Gaussianity: A Misleading Example in the Book Konstantinos Koutroumbas, Sergios Theodoridis - Pattern Recognition
A book reports that ICA cannot be used if the independent components of the analyzed data are Gaussian (at most one can be Gaussian, but no other). However, in the same book, the following example is ...
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Detector for vector-valued signals
I'm trying to find the optimal probability of detection/false alarm for the following detection task. Given $N$ signal samples with $d$ dimensions (independent channels) each, assign the samples to ...
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Remove additive-zero-mean-Gaussian noise
I have 5 values, and each value is corrupted by Additive-Zero-Mean-Gaussian Noise with variance = 0.05.
Each value is in the range from 0 to 1. Is there anyway for me to remove the noise?
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Where does the following expression for stationary Gaussian Noise come from: $\langle \tilde{n}(f)\tilde{n}(f')\rangle = \delta(f-f')\frac{1}{2}S_n$?
First, the definitions:
Definition (Gaussian process)
A random process $X(t)$ is a Gaussian process if for all time points, $t_1,\ldots,t_n$ the random variables $X(t_1 ),\ldots,X(t_n)$ have a ...
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Classifying samples from complex Gaussian distributions
Assume we are trying to classify a sample $X$ as coming from one of two distributions:
$$
\mathcal{CN}(\mu, \sigma^2) \\
\mathcal{CN}(\nu, \sigma^2),
$$
where $\mathcal{CN}$ denotes a (circularly ...
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Confusion about KL divergence between complex Gaussians
The KL divergence between two real-valued Gaussian distributions with means $\mu_1$ and $\mu_2$ and common variance $\sigma^2$ is well known to be:
$$
D_{\text{KL}}\left(\mathcal{N}(\mu_1, \sigma^2) \...
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Understanding y=Hx+n equation in detail?
Consider a wireless communication system having $t$ transmitting antennas and $r$ receiving antennas. Then, the received signal is given by
$y = \mathbb{H}x+n \tag{1}$
where $\mathbb{H}$ is a $r \...
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Bounding Detection and Estimation by SNR in Gaussian Channel
Assume the following problem: A deterministic signal $X$ whose magnitude is known to satisfy $0 \leq \Vert X \Vert_2 < \Delta$ for some known constant $\Delta$ is transmitted through a Gaussian ...
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In this question, what does μ stand for?
Let $X_i$ ($i=1,\ldots,N$) be random variables describing an unknown constant signal $S$ and a noise $n_i$.
$$X_i = \mu_i S + n_i$$
In this question what does "$\mu$" stand for? Also, can I ...
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Why resizing an image smoothed by Gaussian by factor of 2 also increase sigma by factor of 2
In the paper about SIFT algorithm (Distinctive Image Features
from Scale-Invariant Keypoints) it says at the end of section 3.3:
We double the size of the input image using linear interpolation prior ...
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Metric for image sharpness?
Suppose I have a blurry image: a photo convolved with a gaussian blur kernel of unknown sigma. I would want to deconvolve the blurry image using several gaussian kernels (with different sigmas). Is ...
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Probability of the rate of change of a filtered random process
I am given the following problem: I have a filter with impulse response $h(t) = e^{-10t}, t \leq 0$, and autocorrelation function of the input signal, which is WDS and Gaussian with median equal to 0, ...
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How to achieve uniform gaussian profile? Beam profile has grainy/multimode pattern
Hi apologies in advance if this (optics & image-process) is not where I should post this question. Thanks for the help!
I'm required to make a jig that measures the divergence angle of a ...
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FFT of a gaussian signal in Python
I've been trying to get the FFT of a gaussian in Python. When I use the following parameters, the FFT goes hand in hand with the theoretical FT of the gaussian, but if I increase $\sigma$ they rapidly ...
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Gaussian filter: The best parameters for an implementation
Gaussian filter' range is [-inf +inf], but we truncate it for a GMSK modulation implementation.
I have defined the truncated length tr_lg= 1.
...
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Pdf of product of two i.i.d random variable distributed as circular symmetric gaussian
What is a distribution of the product of two circular symmetric Gaussian random variables?
i.e $X ~ CN(0,1)$ and $Y ~ CN(0,1)$ then distribution of $XY.$?
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n-dimentional integral over Multivariate Gaussian
given the prior distribution of $\mathbf{a}=[a_1,\ldots,a_K]^T$ as
\begin{equation}
p_{\mathbf{a}}(\mathbf{a})= \frac{1}{\pi^K \det{\mathbf{R_{\mathbf{a}}}}} e^{\mathbf{a}^H \mathbf{R}_{\mathbf{a}}^{-...
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How to generate wideband Gaussian white noise
I want to generate correlated complex white Gaussian noise signals in MATLAB. What I do is that I take complex Gaussian random variables with unit-variance and multiply them with the desired input ...
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What is the outcome of applying a derivative on X and Y for an image? and Other Gaussian Questions
I've been diving into smoothing kernels and I came up with a lot of questions that I haven't been able to find in the internet. If you can I'd appreciate the help :)
(Capitals and bold were used for ...
user61819
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Are scipy second-order Gaussian derivatives correct?
For an edge detection algorithm, I need to compute second-order derivatives of an image, and I do this with use of Gaussian derivatives. I assumed that the ...
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n independent, normal, random variable distribution
There is 2 time-series signal and we have to compare the distribution of them.
I have heard there is a theory that says for n independent, normal, random variables of a series with many members, the ...
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How is the distribution of the received signal affected if the transmitted signal is changed from complex Gaussian to PSK
Suppose the received signal is $y(n) = h_0s(n)+w(n)$----(1)
Where $h_0, s(n), w(n)$ all are distributed as zero mean complex Gaussian. Thus I know that $y(n)$ will also have the same distribution.
If $...
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When to zero-mean a signal?
I have two sets of signals. The first is a noisy sinewave, which I zero-mean before taking the FFT since I need to find the amplitude.
The other is essentially noise with a gaussian distribution. I'm ...
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understanding the distribution of received signal
In one of the research paper following equations are given :
$y(n) = h_0s(n)+w(n),d = 0$
$y(n) = h_1s(n)+w(n),d = 1$
where $n$ ranges from $1$ to $N$ and
$h_0,h_1$ are wireless channel assumed as $\...
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Why the value of Gaussian curve drop to 1/19 at 2 standard deviation?
Taken from Guide to DSP where it says:
... at two ... standard deviations from the mean, the value of the
Gaussian curve has dropped to about 1/19 ...
It seems to be a straight forward calculation ...
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How to write a complex symmetric gaussian signal [closed]
In my work I need to use this signal of complex symmetric gaussian noise signal represented as $w$. But I don't know exactly how to represent it
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