Questions tagged [gaussian]

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42 views

Does a “chirp”-like generalization of the Gabor or Morlet wavelets definitions exist in the lit somewhere?

I have asked this at the Math SE also. Predicating this on the definition of the continuous Fourier Transform preferred by most electrical engineers: $$ X(f) \triangleq \mathscr{F} \Big\{ x(t) \Big\}...
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2answers
587 views

Fitting a gaussian image using opencv

I'm reading through the opencv documentation and some questions in SO but it doesn't seem to provide this information. I've an image $I(x,y)$ and I want to find a gaussian function $f_{\mu,\Sigma}(x,...
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1answer
58 views

Does a collection of Gaussian random variables necessarily constitute a Gaussian Process?

If $\{X(t)\}$ is a Gaussian Process then the random variables $X(t_k)$ where $k = 1,2,3...n$, are jointly Gaussian. If each random variable $X(t)$ is a Gaussian variable, then will the random ...
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2answers
94 views

Is it a good idea to whiten a colored noise in order to apply formulas for AWGN?

I have posted a question before regarding the detection of a signal under colored Gaussian noise: Energy detection in presence of colored Gaussian noise Theoretical formulas for setting the detection ...
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1answer
36 views

From Uniform to 2D gaussian

I have a 2D uniform distribution. I would like to assign probability-weights in the distribution, according to the standard deviation to convert the distribution into a 2D gaussian. I know how to do ...
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0answers
134 views

Gaussian filter convolution giving unexpected results

I am trying to smooth a time series signal with a Gaussian filter and then differentiate the signal (this is for an application for edge detection). A nice property of convolution is: $$ \frac{d}{dx} \...
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2answers
112 views

MLE parameter estimation — confusion regarding some terms in the pdf of complex normal r.v (Part 2)

This question is based on the application of the pdf which was an earlier question of mine asked here Confusion regarding pdf of circularly symmetric complex gaussian rv If $v \sim CN(0,2\sigma^2_v)$ ...
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2answers
94 views

Derivation of $ R_{N(t)}(\tau) $ from its $f_{N(t)}(\eta)$

How can we prove the auto-correlation function of white gaussian noise $\{ R_{N(t)}(\tau) \}$ is $\frac{N_0}{2} \delta(\tau)$ from its p.d.f in equation below? $$ f_{N(t)}(\eta)=\frac{1}{\sqrt{2 \pi \...
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2answers
238 views

Optimum Filter Signal Detection for Non AWGN Channels

I have been reading this question and it confirms that the matched filter is the maximum-likelihood receiver in the presence of additive white Gaussian noise. So in the AWGN channel it maximizes the ...
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0answers
128 views

Binomial Approximation of Gaussian Distribution

It is said that we may use the binomial coefficients ( a layer from Pascal's triangle) to approximate the 1-D Gaussian kernel with certain $\sigma$, where $\frac{n}{4} = \sigma$ and $n$ is the index ...
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2answers
154 views

Ramp function as derivative in frequency domain?

It is said that to get Laplacian of Gaussian in frequency domain, we may multiply the Fourier transform of Gaussian with two differentiating ramp function (1 ramp gives 1 order of derivative). The ...
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2answers
288 views

Why does second order Gaussian called Laplacian Gaussian?

We usually use Laplacian of Gaussian as the filter for edge detection or blob detection. But the filter itself is essentially a second order Gaussian. So why do we call it Laplacian of Gaussian? Is ...
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1answer
898 views

How to generate random samples of Gaussian distribution directly in the frequency domain?

One can easily draw (pseudo-)random samples from a normal (Gaussian) distribution by using, say, NumPy: ...
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1answer
123 views

Approximating a Gaussian Process

Suppose that $\theta_t$ is an exogenous variable with known Gaussian process. Next, suppose that for any index $i\in [0,1]$, $$ a_{i,t} = (1-\beta)\mathbb E[\theta_t|\mathcal I_{i,t}]+\beta \mathbb E[...
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1answer
273 views

Shannon capacity with distribution different from Gaussian

If I understand correctly, in AWGN channel, for a given SNR, there is a code for $M$ input codeword $\mathbf{X}_{i, 1 \leq i \leq M}$ that $$\lim_{n \to \infty} R=\lim_{n \to \infty}\frac{\log(M)}{n} ...
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2answers
639 views

What is DC level in white gaussian noise?

Am studying unbiased estimators and keep seeing this term "DC Level". What is the expansion of DC and what is a DC Level? Even the Wikipedia page on WGN says nothing about it.
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1answer
81 views

Dimensional analysis of integrated white noise process

This question is somewhat related to this post. Let us consider we have a white noise current source $i_n(t)$, with a variance $\sigma_i^2$, and mean, $\mu_n=0$. Assume that this current is passed ...
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0answers
54 views

Detection over AWGN analysis disagrees with Monte Carlo simulation

I have a fixed vector $\mathbf{s} \in \{\pm1\}^m$ and a random vector $\mathbf{d} \in \{\pm1\}^m$. I send either $\mathbf{x} = \mathbf{x_1} = [\mathbf{s}\space\mathbf{d}]$ or $\mathbf{x} = \mathbf{...
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0answers
159 views

Distribution of the output of AWGN channel

$$\mathbf{Y}=\mathbf{X}+\mathbf{W}$$ $\mathbf{X}$ is input, an $n$-dimension equiprobable vector in a finite subset of complex space $\mathcal{A}^n \subset \mathbb{C}^n$ with norm $\sqrt{\rho n}$ (...
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1answer
483 views

The Distribution of Filtered Gaussian White Noise

Suppose I have some time series $s(t)$ which contains Gaussian white noise generated by a distribution $N(0,\sigma^2)$ Then I apply a filter to s(t) with a frequency response $H(\omega)$, giving me $...
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2answers
1k views

Gaussian filter as a low pass filter

Can anyone explain me how is a Gaussian filter a low pass filter? It may be a simple thing but I just can't seem to wrap my head around it. Also while applying a low pass filter for bandlimiting (to ...
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1answer
201 views

$\tt awgn$ and $\tt wgn$ functions in MATLAB always output the same signal, no matter what I input as SNR

I'm very new to all of this. Right now I'm just trying to add a bunch of different noises to an input signal and see how they affect my signal. This is the current output of my code: ...
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1answer
761 views

How to determine variance/std deviation of Gaussian noise from measured data

If I have a signal measured in discrete time intervals of a process containing some signal and some Gaussian noise how do I go about measuring the variance of this noise [i.e. the $\sigma^2$ term in $\...
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3answers
1k views

On coloured Gaussian noise

It is known that the PSD of additive white Gaussian noise (AWGN) is constant and equal to its variance. What about coloured Gaussian noise (CGN)? For example, given the following PSD of CGN $$S(f) ...
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5answers
837 views

Why does the Kalman filter remove only Gaussian noise?

What and where in the derivation of the Kalman filter is the assumption of Gaussian noise? Why and how does this assumption help?
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1answer
1k views

Is there a mathematical method to determine if noise is Gaussian?

Is there a mathematical method to determine if a signal's noise is Gaussian? The only way I know so far is to analyze the histogram and layover a Gaussian distribution to visually determine if the ...
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1answer
172 views

What is stochastic differential equation and its need?

A white noise process can be simulated using the Matlab command randn(). The numbers will be drawn from a Normal distribution of zero mean and variance 1. Is the ...
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1answer
508 views

Discrepancy between Gaussian FFT and its Fourier Transform

I am trying to do the FFT of a Gaussian signal and comparing it to the theoretical Fourier transform. For infinitely small time step $dt$ and infinitely long signal length $T$, the 2 should become ...
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1answer
1k views

Correct way to add AWGN to a signal

I have a signal, S whose bandwidth is bw Hertz and is sampled at fs Hertz. To contaminate it with a noise corresponding to SNR <...
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1answer
220 views

Interpreting zero-mean Gaussian noise in the context of Kalman Filters

Consider an object moving in 2D space with constant speed $v_x$,$v_y$. If its acceleration is modelled as zero-mean Gaussian $ a_x \sim N(0,\sigma_{a_x}^2)$ and $ a_y \sim N(0,\sigma_{a_y}^2)$ within ...
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2answers
280 views

Why do we express complex noise as Gaussian distributed while the channel is Rayleigh distributed?

I have a confusion regarding the distribution of noise and channel fading. As we know channel fading is usually modeled with Rayleigh fading since it has Gaussian distributed real and imaginary ...
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1answer
87 views

Deriving statistics of band limited Random Noise

The question is: Consider a continuous random number with a Gaussian distribution of mean $\mu$ and variance $\sigma$ . The RV is measured from time $t=-\infty$ to $t=\infty$. This time domain signal ...
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1answer
59 views

Doubts and some confusion on variance for complex rv

This question is in continuation of the one asked here. Let's say that the measurement noise $w$ or any random variable is circularly Gaussian complex. If the imaginary and real components each has ...
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1answer
233 views

Fourier transform of a Gaussian in Excel returns imaginary coefficients [duplicate]

I am using Fourier Analysis tool in Excel to transform a Gaussian $\exp(-x^2)$ on a uniform and symmetric grid of $x$ values. I am expecting the result to contain only real numbers (which should ...
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2answers
7k views

Gaussian noise generation for a given SNR ?

I am trying to add a Gaussian noise, normal distributed to a signal I have simulated (sig_noiseFree), to get a noisy signal (sig_noisy). lease see the code. I wrote the function my self according to ...
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2answers
343 views

Gaussian filter looks pixelated

What does it mean when an image looks pixelated after a Gaussian filter (Multivariate Gaussian Distribution filter) is applied (in context of mu, Sigma, and the meshgrid created), and what parameters ...
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1answer
165 views

Creating a noise model with a fixed PSNR

My goal is to create a noise model so that the PSNR of the noisy image compared to the original is some pre-defined constant $\alpha$. This means the mean squared-error will also be a constant. My ...
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1answer
116 views

Transform coding gain of two channel Haar Wavelet transform (first stage)

Question My interpretation Work I'm attempting to do this in MATLAB. I know that for discrete Random Processes going into an LTI system the autocorrelation function of the output is $R_y[n] =...
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1answer
675 views

Brownian noise is integration of gaussian or uniform white noise?

On this wikipedia page it is written that Brown noise can be produced by integrating white noise What kind of white noise is meant? The Gaussian or uniform white noise?
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2answers
1k views

Parameters of Gaussian Kernel in the Context of Image Convolution

Hi Everyone i am new at image processing. I copy code from this website, I have image $600 \times 480$ gray scale. What will be the value of standard deviation or $\sigma$? What will be the value of ...
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3answers
603 views

Capacity of AWGN channel

I am confused understanding basic concepts of communication over AWGN channels. I know the capacity of a discrete time AWGN channel is: $$C=\frac{1}{2}\log_2\left(1+\frac{S}{N}\right)$$ and it is ...
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1answer
204 views

Parameters of Butterworth filter that approximates the amplitude response of a Gaussian filter?

For a given narrowband Gaussian filter with a specific central frequency and filter width, I need corners of a bandpass Butterworth filter whose amplitude response is close enough to the Gaussian ...
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1answer
702 views

Confusion regarding pdf of circularly symmetric complex gaussian rv

Considering a random variable $x$ that takes in values from a complex domain. Its real and imaginary components are totally uncorrelated. I am following this link and also studying this document. In ...
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2answers
687 views

bandwidth limited pulse,what window function is it?

I want to know what kind of shape a multiple cycle sine wave pulse ( or maybe the correct term is burst ) should have to have the narrowest bandwidth possible at given pulse width. In the laser ...
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1answer
553 views

Value of power spectral density $N_0$ or effect of scaling bandwidth to SNR

I want to compare performance over AWGN channel of two systems that are exactly identical but their bandwidth. I will talk only about baseband system for sake of simplicity. For more details about ...
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3answers
304 views

Is there any way to measure of Gaussian-ness?

I have some sampled data that has $1/f$ noise in it, with departures from the mean. These are long term departures. I could use something like a median filter but the window length would be longer ...
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1answer
966 views

Convolution of Two Kernels [OpenCV]

I've been completely stuck on a portion of my assignment for a few days now. After plenty of searching around, I have been unsuccessful in discovering information that leads me to the correct solution....
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1answer
265 views

What should be the output of the pdf for multivariate normal distrubition?

Let's think we have a multivariate normal distribution with $$\mathbf x_1=\begin{bmatrix}x_{11}\\x_{12}\end{bmatrix},\quad \mathbf x_2=\begin{bmatrix}x_{21}\\x_{22}\end{bmatrix},\quad\mathbf m=\begin{...
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2answers
207 views

Additive white Gaussian noise and undecimated DWT

One of the benefits of DWT is that it is an orthonormal transform. There are statements that the energy of noise component mainly concentrates on the high-frequency (detail) part and distributes ...
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1answer
2k views

Why does the separable filter reduce the cost of computing the operator?

A separable filter in image processing can be written as product of two more simple filters. Typically a 2-dimensional convolution operation is separated into 2 onedimensional filters. This reduces ...