Questions tagged [noise]

This tag is for questions about noise. In signal processing, noise is a general term for unwanted (and, in general, unknown) modifications that a signal may suffer during capture, storage, transmission, processing, or conversion.

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Can I calculate anything useful from equation assuming regarding noise

I have a setup where three measurements are made from arbitrary positions, which then have to be put together to create corresponding values in a orthonormal coordinate system. The formula has been ...
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36 views

Variance of the Kalman filtering

Having the scalar system: \begin{cases} x(t+1)=ax(t)+b\eta(t)\\ y(t)=cx(t)+d\xi(t) \end{cases} where $\eta(t)=\text{WN}(0,1)$ and $\xi(t)=\text{WN}(0,1)$ are uncorrelated noises. Why ...
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8 views

Coherent Noise function producing dissimilar results for similar inputs

Is there a noise function I(x,y) which produces dissimilar output for similar inputs? The primary purpose of it would be to mask coordinates so any point could have ...
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17 views

Mathematical Definition of White Noise

I wish to understand more about Gaussian white noise through a formal math definition if possible and how its related to the mean and variance (if we are speaking in the time-continuous domain) or if ...
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50 views

Variance of white noise integration

So I know that the variance and mean of white noise is $$ E[X] = 0$$ $$ Var[X] = \frac{No}{2}$$ So what would the variance and mean be for the follwong equation $$ S = \frac{1}{T} \int_0^T{x} dt$$ ...
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39 views

When can standard numerical integration techniques be used to integrate a system with noise?

I know that in order to solve a differential equation with white noise a stochastic numerical integration technique must be used. As I understand it, this is because the white noise has zero ...
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14 views

Adding noise in the frequency domain when computing an SPDE with Fourier Transforms

currently I'm trying to implement a Galerkin method to solve a certain kind of stochastic PDE in 2D with additive noise. Using the Karhunen-Loeve series expansion we can build a Galerkin method to ...
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1answer
28 views

Proving independence when Gaussian noise is added

Consider r.v.'s $X$ and $Y$ which are not independent, i.e. $E[X\mid Y]\neq E[X]$. Can we show that if we add a variable $\epsilon$ following $\mathcal N(0,\sigma_{\epsilon})$, for which it holds that ...
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1answer
37 views

Correct interpretation of noise-based wind field generation by Khorloo Oyundolgor

I am interstring in the creation 3D wind field for my falling snow scene using DirectX. After learning different approaches, I came to a conclusion to use the following method because of the detailed ...
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43 views

Divergence when calculating moments with a space-time Gaussian white noise

Consider a variable $M(x,t)$ driven by space-time Gaussian white noise: $$ \partial_t M(x,t) = -k M + \xi(x,t).$$ The noise has mean $\langle \xi(x,t) \rangle = 0$ and correlation function $\langle \...
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1answer
22 views

Improving system identification results?

I'm performing system identification of the lateral closed-loop dynamics of a quadrotor. My model receives a setpoint and should return position and acceleration. I've proposed a second order model of ...
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43 views

Average probability of detection error with Gaussian noise vector

I have a problem with how to express the average probability of error when The Probability involves vectors. The problem is a classic detection problem where the probability under the Hypothesis $S_{k}...
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1answer
44 views

Newton Raphson for nonlinear overdetermined equations on noisy data [closed]

I would like to know, whether any improved Newton Raphson method is available for non-linear overdetermined equations (So we use Jacobian matrix and pseudo inverse). Data used as measurements are ...
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1answer
32 views

Estimation from noisy observation

Assume that you have access to a noisy observation (static vector) y, which can be expressed as $y=x+b$ where the static vector $x$ is unknown. The noise $b$ is an independent and identically ...
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28 views

Is there a direct connection between Markov processes and white noise processes?

I am trying to establish a mathematical foundation that shows whether a stochastic process (s.p.) $X=(X_t)_{t\geq0}$ is a Markov process if and only if its mean-square derivative, $\dot X_t=\frac{d X}{...
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1answer
40 views

Variance of ARMA process?

Problem Given $ARMA(1,1)$ stationary process $$x_t = 0.7 x_{t-1} + u_t + 0.2 u_{t-1} $$ where $u_t$ is white noise, with standard deviation $\sigma(u_t) = 4$ Note, stationarity of $x_t$ implies that $$...
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24 views

Derivative of noised function

Given a noisy smooth enough function, $f(x)=sin(x)$ find the derivative using the sliding derivative technique. In the sliding derivative technique, we are performing the following steps: The ...
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10 views

Derivative definition is stochastic systems

Assume that we have the following stochastic dynamical system: $$dx=f(x,p)dt+\sigma dW_1$$ and another system at $p+\delta p$ as: $$dx'=f(x',p+\delta p)dt+\sigma dW_2$$ Now I want to write down an ...
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19 views

Showing weak lower semi-continuity on a functional containing white noise

We know from the book of Evans: PDEs, that if the Langratian $L(p,z,x)$ is bounded below and the function $p \mapsto L(p,z,x)$ is convex, for all $z \in \mathbb{R}$, $x\in U$, then the corresponding ...
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13 views

Noise Effect on Autoregressive Polynomial Roots

So suppose I have the following signal: $$ x(t) = s(t) + n(t) $$ where $n(t) \sim \mathcal{N}(0, \sigma_{n}^{2})$ and $s(t)$ is an autoregressive process of order $P$. My question -- If I model $x(t)$ ...
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66 views

Solve Langevin equation with blue noise?

I would like to solve the following Langevin equation $$\frac{d^2 x}{d t^2}+\omega_0^2x(t)=\eta(t),$$ where $\eta(t)$ is a blue noise signal given by $$\eta(t)=\int_{-\infty}^\infty \hat{\eta}(f)\exp(...
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1answer
73 views

Confirming solution to a Langevin equation using Fourier series

Consider the following Langevin equation $$\frac{d^2 x}{dt^2}+\omega_n^2x=\eta(t),$$ where $\eta(t)$ has a gaussian probability distribution with mean zero and correlation $$\langle \eta(t) \eta(t')\...
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1answer
141 views

Poisson equation with stochastic source

In a physical set-up, one can consider an electrostatic problem where the charge density at each point in space is a random variable, and try to find the electric potential or electric field. To be ...
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39 views

Results on Reconstruction Error in Random Matrix Thoery

I am curious about the current state of random matrix theory as pertains to the following question. I spent about an hour searching through literature and struggled to find material that directly ...
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1answer
53 views

Simulating random movememt

I'm trying to simulate random movement in an particle caused by temperature, by adding a random vector of length $1$ every simulation step. The problem is that as I decrease the step time, the average ...
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50 views

Use of least-squares minimization to solve noisy systems of linear equations

This question is somewhat connected to a previous one I posted two months ago concerning solving linear, over-determined systems of equations of the form $Ax = b$, where $A$ is a matrix of ...
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1answer
17 views

Estimation of squared normal distribution

I am given a $w \sim N(0,I_n)$ and $w \in \mathbb{R}^n$ and $X \in \mathbb{R}^{n \times d}$ such that $X_1,..., X_d \in \mathbb{R}^n $ of $X$ that satisfy $\|X_i\|^2 = n$ where $n$ is a scalar and ...
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1answer
154 views

Can a Wiener process be obtained as the limit of a “memoryless collision time” model?

Let $(N_t)_{t \geq 0}$ be a Poisson process of intensity $1$, and for each $\lambda>0$ and $t \geq 0$ let $$ W^{(\lambda)}_t = \sqrt{\lambda} \int_0^t (-1)^{N_{\lambda s}} \, ds = \frac{1}{\sqrt{\...
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313 views

Probability of a non-random sample to represent initial Poisson distribution

I'm looking for a way to correctly approach the following noise-signal problem. What I'm doing is a looking for arbitrary structures in a seemingly random input data, akin to a night vision with noisy ...
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228 views

Solving 'noisy' systems of equations

My question is concerned about solving linear, over-determined systems of equations of the form $Ax = b$, where $A$ is a matrix of coefficients of dimension $m \times k$, $(m > k)$, $x$ is the ...
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23 views

Is Noise Figure dependent on input noise power?

I was reading about the Noise Figure on Wikipedia, where I saw the following definition: The noise factor is thus the ratio of actual output noise to that which would remain if the device itself ...
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1answer
35 views

Need help with simplex noise skew transformation

I am reading a paper about simplex noise. http://knielsen-hq.org/simplex_noise_skew_factor.pdf For whatever reason I can't figure out the result they got here. My brain is just goin kapoot. To ...
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34 views

How to schematize a noise signal in probability theory

I have never studied random processes, but in my baggage I carry only basic concepts of probability that can all be enclosed in the first chapter of this book https://www.springer.com/gp/book/...
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Does A Continuous Time White Noise Process Actually Exist?

I have seen white noise defined as a zero-mean stochastic process with the following autocorrelation function (in this question, for example Time continuous white noise): \begin{align*} E[X(t)]E[X(t+\...
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1answer
35 views

White noise not strongly stationary

Let $t=0,1,2,3,...$ a discrete time index. Consider the following process where $\gamma(t_1,t_2)$ indicates the autocovariance function. I want to prove that the process (although being ...
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1answer
66 views

Why mathematically can the noise $\epsilon$ of a regression be dropped when computing the density $p(y|x, \theta)$

In the notes of Andrew Ng on linear regression p.11-12 it is written the following: Let us assume that the target variables and the inputs are related via the equation $$ y^{(i)} = \theta^\top x^{(i)}...
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25 views

Hurst exponent of fractional gaussian noise

Given a fBm $B_H(t)$, we define the fGn $G(j)$ as $$ G(j) = B_H(j+1)-B_H(j) \quad j\in\mathbb{N} \ . $$ If $H=\frac12$, then $B_H(t)$ is the usual brownian motion, and $G(j)$ is a (discrete) white ...
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1answer
36 views

Calculate SNR if I have the noise?

I am looking for ways to calculate signal-to-noise ratio (SNR). As I understand it, this measure is often used when you have a separated clean signal and noisy signal, and can thus measure the power ...
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201 views

Ito Derivative of White Noise

We know that white noise $w_{t}$ is given by the time derivative of Brownian motion $\beta_{t}$, ie that: $$ w_{t} = \frac{d \beta_{t}}{dt} $$ Now I want to define a new process, called blue noise $...
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1answer
35 views

Examples of Multiplicative Noise

I understand the main idea behind a multiplicative noise in signal processing, but I'm struggling to see it expressed in a specific example. Could someone help me? For example, if I have a system of ...
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34 views

Question about the limit of the variance of a band-pass limited white noise

Let $s(t)$ be a Gaussian process with zero mean, unit variance and flat power spectrum given by $ S_{ss}(f)=\frac{1}{2(f_2-f_1)} $ for $ \vert f \vert \in [f_1,f_2]$, where $ f_1 $ and $f_2$ are ...
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2answers
343 views

What is the difference between Noise and chaos?

Two terms are mixed me , I heard about noise from stochastic process phenomena and i heard about chaos from dynamical system , then Is there someone who can help me to get difference between noise and ...
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2answers
201 views

What is the point of using “white noise” term in maths?

I've been studying on my master thesis about "Stochastic Differential Equations". Since I'm new to this topic, I couldn't understand the relation between "white noise" and mathematics. I searched ...
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1answer
169 views

Do i.i.d. stochastic processes exist in continuous time?

Does there exist a stochastic process $X_t$, $t \in [0,\infty)$, such that $X_t$ is distributed according to some distribution $f(x)$ that possesses finite variance and such that $X_t$ and $X_s$ are ...
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1answer
44 views

White noise RMS vs. its bandwidth

From numerical simulation and regression analysis I discovered that the root-mean-square amplitude of white noise with bandwidth $\Delta\!f$ is proportional to $\sqrt{\!\Delta\!f}$. How can this be ...
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65 views

Would sampling the decimal digits of $\pi$ generate a white noise signal?

Discrete r.v. $X = \pi(d)$ (defined in another q of mine). Discrete r.v. $Y = X - 4.5$. q1: Would it be incorrect to deduce $Y\sim U(-4.5,4.5)$ from $X\sim U(0,9)$? q2: If you answered no to q1, ...
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1answer
165 views

How is the noise gain function defined for higher order discrete piecewise white noise in a Newtonian system?

Background I have been trying to understand Kalman filters and implement them in a project I have. I have been following Roger Labbe's online book (https://nbviewer.jupyter.org/github/rlabbe/Kalman-...
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1answer
84 views

Upper and Lower shapes in Perlin Noise Generated Terrain

I am trying to learn about Perlin Noise and procedural generation. I am reading through an online tutorial about generating landscapes with noise, but I don't understand part of the author's ...
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1answer
110 views

Unit used in continuous time process noise matrix in kalman filters, when STD is from discrete time data

I'm trying to make a process noise matrix in continuous time. But i can't seem to find a clear definition of what "unit" the matrix should contain in continuous time. From our control book we have $...
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321 views

Cross-correlation of a deterministic signal and white Gaussian noise

I'm trying to describe the cross-correlation of a finite length input signal x[n] with the same signal corrupted by white Gaussian noise. If the signal would be infinitely long, the noise would be ...