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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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
20 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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39 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
37 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
29 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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24 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
30 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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23 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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9 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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16 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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11 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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61 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
62 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
129 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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36 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
51 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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40 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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21 views

PSD from Wiener Khintchine Theorem

i have a question regarding the Wiener Khintchine Theorem. So as far as i understand $S_{xx}(f) = \mathcal{F}\{R_{xx}(t') \} $. $R_{xx}(t') = \mathbf{E} \{x(t)x(t+t')\} $ Where S ist the PSD and R the ...
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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
131 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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310 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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155 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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13 views

How do you determine noise immunity of Pulse Position Modulation or Pulse Amplitude Modulation?

This website has the following table: It says the noise immunity for PAM, PWM, and PPM is low, high, and high respectively. How does one compute this noise immunity? My thinking is for each of ...
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21 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
33 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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31 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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76 views

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
31 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
59 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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23 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
31 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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155 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
26 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
240 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
179 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
147 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
34 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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61 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
142 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
74 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
82 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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263 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 ...
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0answers
40 views

gaussian white noise implies gaussian arma process

An ARMA(p,q) process is a (weakly) stationary process $x_t=\sum_{i=1}^p\phi_ix_{t-i}+z_t+\sum_{j=1}^q\theta_j z_{t-j}$ where $z_t$ is white noise. Lets assume that $z_t$ is Gaussian white noise. ...
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1answer
34 views

Determine stationarity of time series containing sin of white noise [closed]

Could someone help me determine the stationarity of the the following time series Y? $ Z_t $ represents white noise with variance $ \sigma^2 $. $ Y_t = \sin(Z_t) + Z^2_t - Z_{t-1}$ I have tried ...
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122 views

Autocorrelation in a Brownian model

I have the following brownian model: $$ \dot{x}=v_0cos(\theta(t))+\sqrt{2D_t}\xi_x(t) \\ \dot{y}=v_0sin(\theta(t))+\sqrt{2D_t}\xi_y(t) \\ \dot{\theta}=\sqrt{2D_r}\xi_\theta(t)\\ $$ with $v_0$ ...
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0answers
29 views

Simulating a random process with AWGN

I am trying to simulate the following random process: $$ \frac{dy}{dt} = f(y) + AN(t) $$ where $N(t)$ is additive Gaussian white noise, which, if I remember right, is defined by $\int\limits_0^t N(t)...
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1answer
382 views

How to code oscillator driven by Gaussian white noise? Edit: How to convert ODE to a system of SDE's?

I have written some python code which was designed to try to solve the following differential equation: $$\ddot{x}+\omega_0^2x=\eta(t),$$ where $\eta(t)$ is the gaussian white noise, with mean 0 and ...
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39 views

$E(X(t)X(t))=\sigma^2\delta(\tau)$ or $E(X(t)X(t))=\sigma^2$ [duplicate]

Let's say we have a white noise process $x(t)$ such that: $E(X(t)X(t+\tau))=N\delta(\tau)$ $E(X(t))=0$ In particular, with $\tau=0$, $E(X(t)X(t))=E(X^2(t))$ is infinite. Now, I want $X(t)$ at each ...
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2answers
552 views

Will the energy of a randomly driven harmonic oscillator grow to infinity or oscillate about a finite value?

The equation of motion for an undamped harmonic oscillator, with driver $f=f(t)$ is given by: $$\ddot{x}+x=f.$$ Let the initial conditions be given by: $$x(0)=\dot{x}(0)=0.$$ If $f=\cos(t)$ ...