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.
Questions tagged [noise]
234 questions
54
votes
2 answers
What is meant by a continuous-time white noise process?
What is meant by a continuous-time white noise process?
In a discussion following a question a few months ago, I stated that as an engineer, I am used to thinking
of a continuous-time wide-sense-stationary white noise process
$\{X(t) \colon…
Dilip Sarwate
- 26,411
19
votes
4 answers
Why is gradient noise better quality than value noise?
I have been reading about the mathematics behind Perlin noise, a gradient noise function often used in computer graphics, from Ken Perlin's presentation and Matt Zucker's FAQ.
I understand that each grid point, $X$, has a pseudo-random gradient…
Joseph Mansfield
- 293
14
votes
2 answers
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 $f=\cos(t)$ then the solution is:
…
Peanutlex
- 1,169
8
votes
1 answer
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 $b_{t}$, and define it as:
$$ b_{t} = \frac{d…
the_src_dude
- 313
7
votes
1 answer
Can a Wiener process be obtained as the limit of a "memoryless collision time" model?
Let $(N_t)_{t \geq 0}$ be a Poisson process of intensity $1$, and for each $\lambda>0$ and $t \geq 0$ let
$$ W^{(\lambda)}_t = \sqrt{\lambda} \int_0^t (-1)^{N_{\lambda s}} \, ds = \frac{1}{\sqrt{\lambda}} \int_0^{\lambda t} (-1)^{N_s} \, ds. $$
Is…
Julian Newman
- 519
6
votes
0 answers
Does A Continuous Time White Noise Process Actually Exist?
I have seen white noise defined as a zero-mean stochastic process with the following autocorrelation function (in this question, for example Time continuous white noise):
\begin{align*}
E[X(t)]E[X(t+\tau)] = \begin{cases}
\sigma^2, \tau=0…
gigalord
- 337
5
votes
1 answer
How to numerically solve noisy ODE
I need to numerically integrate a large number of ODE's of the following form
$$
\dot{X} = k_{1}\left[\rule{0pt}{4mm}U\left(t\right) - X\right] + k_{2}V\left(t\right)\left[\rule{0pt}{4mm}W\left(t\right) - X\right]
$$
Here
All variables and…
Aleksejs Fomins
- 2,348
5
votes
1 answer
Is the integral of the Dirac delta function equal to the integral of the Dirac delta function times the Heavisde unit step function?
Given that the Dirac delta function is defined as:
$$
\delta(t) =
\begin{cases}
+\infty, & t = 0\\[2ex]
0, & t \neq 0\\[2ex]
\end{cases}
$$
And that the Heaviside unit step function is defined as:
$$
\Theta(t) =
\begin{cases}
0, & t <…
JMy
- 51
5
votes
1 answer
Interpretation of "Noise" in Function Optimization
I am trying to better understand the meaning of "noise" with regards to function optimization - specifically, why "Noisy" functions are more difficult to optimize compared to "Non-Noisy" functions.
Up until now, I always thought of "noise" from a…
stats_noob
- 4,107
5
votes
2 answers
Scaling of space-time white noise
On different sources I found different parabolic scalings for space time white noise that I believe are in contradicton one with the other.
Let $\xi(t,x)$ be space-time white noise on $\mathbb{R}\times\mathbb{R}^d$. I apply a scaling $t\to…
pollastro
- 51
4
votes
0 answers
What is the math symbol ~ with ind over it?
The symbol I'm talking about is from a statistics article here:
EwokNightmares
- 141
4
votes
2 answers
Given a Poisson-noisy signal, what is the noise distribution of its Fourier transform?
Disclaimer: I'm not a mathematician, but here's my attempt at a mathy version of my question
Start with a noiseless, discretely sampled expected signal $I(x_n)$. Construct a Poisson-noisy measurement of this signal $P(I(x_n))$, by drawing samples…
Andrew
- 307
4
votes
3 answers
How to dither binned data (following a geometric distribution) to recover the exponential distribution?
Consider a random variable that follows an exponential distribution. After binning (floor), becomes discrete and follows a geometric distribution. My question is: how can we recover the original exponential distribution by adding random noise to…
Roland
- 143
- 6
4
votes
1 answer
Whiteness hypothesis in Kalman filtering
In Kalman filter mathematical treatment I have always read that a foundamental hypothesis is represented by the whiteness of the process noise.
I have tried to do again the mathematical steps in the Kalman filter derivation but I can't see where…
Nameless
- 89
4
votes
1 answer
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 independent for all $s \neq t$?
Unlike other…
user3353819
- 483